SIMULATION OF SODIUM BOILING EXPERIMENTS
WITH THERMIT SODIUM VERSION
by
Kang Yul Huh
MIT-EL 82-023
May 1982
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SIMULATION OF SODIUM BOILING EXPERIMENTS
WITH THERMIT SODIUM VERSION
by
KANG YUL HUH
B.S., Seoul National University
(1980)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF
MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 1982
-2-
SIMULATION OF SODIUM BOILING EXPERIMENTS
WITH THERMIT SODIUM VERSION
by
KANG YUL HUH
Submitted to the Department of Nuclear Engineering on
May 7, 1982, in partial fulfillment of the requirements
for the Degree of Master of Science
ABSTRACT
Natural and forced convection experiments(SBTF and French)
are simulated with the sodium version of the thermal-hydraulic
computer code THERMIT. Simulation is done for the test secti-on with the pressure-velocity boundary condition and subsequ-ently extended to the whole loop. For the test section simu-lation, a s.teady-state and transient calculations are perform-ed and compared with experimental data. For the loop simula-tion, two methods are used, a simulated 1-D loop and an actual
1-D loop. In the simulated 1-D loop analysis, the vapor densi-ty is increased by one hundred and two hundred times to avoid
the code failure and the results still showed some of the impor-tant characteristics of the two-phase flow oscillation in a
loop. A mathematical model is suggested for the two-phase
flow oscillation. In the actual 1-D loop, only the single
phase calculation was performed and turned out to be nearly the
same as the simulated 1-D loop single phase results.
In the process of simulation, it is discovered that the
energy conservation equations of THERMIT fails to conserve
energy by a small amount due to interfacial mass transfer and
frictional dissipation of mechanical into thermal energy. The
problems and applicability of the THERMIT physical models are
also discussed.
Thesis Supervisor: Neil F. Todreas
Professor of Nuclear Engineering Department
-3-
ACKNOWLEDGEMENTS
I wish to express my sincere gratitude to my thesis
supervisor Professor Neil E. Todreas for his guidance and
advice, and to Professor Mujid S. Kazimi as my thesis reader.
Finally I will never forget Andrei L. Schor for his friend-ship and advice throughout all the difficulties I encoun-tered for the last sixteen months.
-4-
TABLE OF CONTENTS
page
Abstract. ...............................................
2
List of Figures .........................
.........
8
List of Tables ..........................
.eoooeeoeoeee
Nomenclature............................
.ooeoe
11
oo
Chapter 1:
Introduction................
......
Chapter 2:
Simulation of the cases.....
ooeooeoooeoooo.
2.1
Natural convection................
o...
12
oooo
16
o........
oooo
o
21
o
21
oooo.
2.1.1
Explanation of the experiment ORNL/TM-7018.. 21
2.1.2
Calibration of heat input....
oeeeoooeeoetoee
24
2.1.3
Test section simulation......
oooeooooooooo
26
...............
26
2.1.3.1
Steady-state............
2.1.3.1.1
Boundary conditions..
2.1.3.1.2
Geometry.............
2.1.3.2
2.1.4
Transient ................
o
.o
oo
o
o
oo
ooeo
o
o
o
27
o
...............
oeooeo
.....
oooo
o....e.....
oo
oooooo
.
Geometry and boundary con ditions of
the simulated 1-D loop... .. ....... e..00...
2.1.4.2
. ..
o. ..
. .
.
.o.o
37
Geometry and boundary con ditions of the
actual 1-D loop ......
2.2
32
Artificial vapor density in the simulated
1-D loop.............
2.1.4.3
31
32
Whole loop simulation........
2.1.4.1
26
Forced convection............
38
eeeeeeee.,oooo,
...
oeooo
ooeooo
o
42
-5-
2.2.1
Explanation of the experiment ...............
42
2.2.2
Geometry and boundary conditions ............
42
Chapter 3:
3.1
Numerical and physical models............... 46
Numerical models. ...............................
46
THERMIT conservation equations ..............
46
3.1.1
3.1.1.1
HEM, 4-Eq and 6-Eq models............... 46
3.1.1.2
Energy loss due to dissipation and
interfacial mass transfer................ 51
3.1.2
Explanation of the code from numerical
point of view................................ 53
3.2
....... .......................
61
Physical models in THERMIT..................
61
Physical models..
3.2.1
61
3.2.1.1
Wall friction............................
3.2.1.2
Interfacial momentum transfer........... 64
3.2.1.3
Wall heat transfer......................
3.2.1.4
Interfacial heat transfer............... 67
3.2.1.5
Interfacial mass transfer ............... 67
3.2.2
64
Problems of THERMIT physical models......... 69
3.2.2.1
Forced and natural convection............ 69
3.2.2.2
Flow regime ............................. 69
3.2.2.3
Condensation modeling ...................
3.2.3
72
Translation of physical models to different
geometry .................................... 73
3.2.3.1
Equivalent hydraulic diameter........... 73
-6-
3. .3.2
Summary of physical models for circular
3.2.
tube geometry...................... .........
76
Results ........................... .........
80
Natural convection ..................... .........
80
Test section simulation ........... .........
80
Chap ter 4:
4. 1
4.1.1
4.1.1.1
Steady-state ................... .........
80
4.1.1.2
Transient...................... .........
89
4.1.2
4. 2
Whole loop simulation.............. ......... 89
4.1.2.1
Simulated 1-D loop analysis.... .........
4.1.2.2
Actual 1-D loop analysis.......
89
......... 98
Forced convection ...................... .........
111
Discussion........................ .........
116
Natural convection..................... .........
116
Test section simulation............ .........
116
Steady-state ................... .........
116
Chap ter 5:
5. 1
Interfacial area......
5.1.1
5.1.1.1
5.1.1.1.1
Pressure drop.............. ......... 116
5.1.1.1.2
Energy conservation........ .........
5.1i.1.3
Comparison of HEM, 4-Eq and 6-Eq
119
models ..................... ......... 120
Transient ...................... .........
121
Whole loop simulation.............. .........
122
5.1.1.2
5.1.2
5.1.2.1
Oscillation.................... .. ....... 122
5.1.2.2
Mathematical model of oscillation....... 123
-7-
Chapter 6:
Conclusions...................... ..........
Chapter 7:
Recommendations for future work.. ..
References.
.............................
•..
..
134
.....
136
.......... 138
Appendix A:
Calibration of heat input....... ..........
Appendix B:
Energy conservation of THERMIT.. ......... .144
Appendix C:
Plenum temperature ..............
Appendix D:
Dependence of the wall friction
140
..148
void
fraction.........................
152
Appendix E:
Typical experimental data.......
155
Appendix F:
Typical computer inputs and outputs.......162
-8-
LIST OF FIGURES
Number
Page
2.1
Sodium Boiling Test Facility - loop............... 22
2.2
Sodium Boiling Test Facility - test section....... 23
2.3
Geometry for the test section simulation......... 29
2.4
I
2.6
Schematic diagram for geometry transformation....
2.7
Typical experimental data for steady-state boiling
30
experiment ........................................ 33
2.8
Geometry for the simulated 1-D loop..............
2.9
Geometry for the code inputs of the simulated 1-D
34
loop...................................................................... 35
2.10
Geometry for the actual 1-D loop.................. 39
2.11
Geometry of the actual 1-D loop analysis......... 40
2.12
French experimental facility...................... 43
2.13
Geometry of the hypothetical forced-convection
French experiment.... .........................
3.1
Typical velocity profiles for forced and natural
convection.................................
3.2
44
70
Interfacial area for rod-bundle and circular tube
geometry with the same equivalent hydraulic dia-meter..-.....
4.1
..................
...........
Pressure distribution for the test run 129R1.....
77
84
-9-
4.2
Oscillation of the pressure drop according to the
inlet velocity for test run 129R1................ 90
4.3
Oscillation of the void fraction, mixture enthalpy
and mixture density at the end of the heated zone
91
for 129R1 ........................................
4.4
Oscillation of liquid and vapor velocities at the
92
outlet for the transient of the test run 129R1...
4.5
Pressure drop over the test section for simulated
93
1-D loop analysis with vapor density x200........
4.6
Inlet velocity of the test section for simulated
1-D loop analysis with vapor density x200........ 94
4.7
Pressure drop over the test section for simulated
1-D loop analysis with vapor density x100........ 95
4.8
Inlet velocity of the test section for simulated
96
1-D loop analysis with vapor density xl00........
4.9
Stable boiling oscillation in the simulated 1-D loop
with vapor density x200 ..................... .....
99
4.10
Spatial void distribution for simulated 1-D loop
with vapor density x200..................... .....
100
4.19
4.20
pressure distribution for single-phase loop ana-lysis for the test run 107R2.................... 110
4.21
S-curve for two different flow areas for the hypo-thetical French experiment ...................... 114
5.1
Schematic diagram of the loop and the test section.124
-10-
5.2
Return loop.......................................129
5.3
Calculation results that show the relationship
between P -P
Al
and V -V .......................... 133
Relative error in test section power determination
as a function of test section power..............142
A2
Comparison of the actual power and the reported
power in ORNL/TM-7018............................143
Cl
Plenum energy conservation. ...................... 148
C2
Iterative scheme..................................150
El
Stable boiling of the test run 129R1 at 1.3 kW...156
E2
Stable boiling of the test run 127R1 at 1.5 kW...157
E3
Void detection system data for the test run 127R1.158
E4
Relationship between voiding pattern, flow and
pressure for the test run 127R1 .................. 159
E5
Stable boiling at high flow and low quality for
the test run 120R1 at 1.5 kW.....................160
E6
Relationship between voiding pattern, flow and
pressure for the test' run 120R1...................161
-11-
LIST OF TABLES
Page
Number
2.1
SBTF natural convection results.................... 25
3.1
Summary of the constitutive relations used for the
natural and forced convection simulation..........
4.1
Steady-state test section pressure drop for 4-Eq/
6-Eq models .........
4.2
79
..............................
82
Steady-state test section calculation convergence
criteria for 4-Eq model........................... 83
4.3
Steady-state test section energy conservation for
4-Eq model........................................ 86
4.4
Steady-state HEM model for test run 129R1.........
87
4.5
Steady-state 4-Eq/6-Eq model for test run 129RI...
87
4.6
Dependence of the pressure drop on interfacial
friction for test run 129R1 .......................
88
4.7
Forced convection calculations for r = 0.002 m....112
4.8
Forced convection calculations for r = 0.003 m....113
D1
Wall friction forces by the liquid and vapor phases
in contact with the wall.......................... 154
-12-
NOMENCLATURE
Definition
Letter
A
flow area in axial direction
At
true interfacial area between liquid and vapor
cf£
wall contact fraction of liquid
cfv
wall contact fraction of vapor
c
specific heat at constant pressure
D
diameter of the tube
De
equivalent hydraulic diameter
e
internal energy per unit mass
f
wall friction factor
F
friction force
g
gravity
h
enthalpy per unit mass
H
wire wrap lead of the helical spacer
hfg
enthalpy change between liquid and vapor at
saturation
h£fc
-
wall heat transfer coefficient through liquid
convection
hinb
wall heat transfer coefficient through nucl-eate boiling
hvfc
wall heat transfer coefficient through vapor
convection
k
thermal conductivity
-13-
L
axial length
mass flow rate
M
mass
M
geometry factor for wall friction calculation
P
pressure
P
pitch between adjacent rods
P
pressure at the top face of the lower plenum
P'
1
pressure at the bottom face of the lower
plenum
P0
pressure at the top face of the upper plenum
PI
pressure at the bottom face of the upper
plenum
Pe
Peclet number
Pr
Prandtl number
q
heat transfer rate per unit area
Q
heat transfer
Re
Reynold's number
R
gas constant of sodium
S
slip ratio
t
time
T
temperature
tz
liquid temperature
tsat
saturation temperature
tv
vapor temperature
u
velocity
rate per unit volume
-14-
v
velocity
V
volume
Vfg
specific volume change between liquid and
vapor at saturation
flow quality
axial coordinate
Greek
void fraction
interfacial mass transfer rate per unit volume
interfacial mass transfer coefficient
viscosity
P
density
a
surface tension
Superscript
n
time step at n
Subscipt
c
condensation
dry
dryout
e
evaporation
i(in, inlet) inlet
k
spatial point at k
liquid
-15-
lam
laminar
m
mixture of liquid and vapor
0(out)
outlet
r
return loop
s
(used with T) saturation
s
(used with p, e, u, V) single-phase liquid
T
total of the liquid and vapor contributions
trans
transition between laminar and turbulent
turb
turbulent
v
vapor
w
wall
z
axial coordinate
II
--16-
Chapter 1.
Introduction
This thesis is primarily concerned with applying the
computer code THERMIT
(sodium version) to actual experiments
to examine the applicability of the numerical and physical
models in the code by comparing calculational results with
experimental data.
THE.RMIT is a component code for thermal-hydraulic
calculations of a nuclear reactor core.
It was originally
written for water coolant, and revised for use in LMFBR's
by
-AndreiL.
Schor at MIT.
Schor has also prepared the
4-Eq model as part of the sodium version of THERMIT so that
there are now three two-phase flow models, .i.e. a 6-Eq,
a 4-Eq and a HEM model.
The THERMIT version that is used
throughout this thesis is the Schor version (as of January,
1981) which has all the three two-phase flow models as
options.
The basic THERMIT code-uses the two-fluid model with
mass, momentum and energy conservation equations for each
phase, hence six conservation equations.
Presently the
constitutive relations of the two fluid model are not in a
well-established state so that in some cases it
may be better
to use the 4-Eq model than the 6-Eq model.
The 4-Eq model has mixture mass and energy conservation
-17-
equations in addition to a separate momentum conservation
equation for each phase.
Physically this equation set
requires that the liquid and vapor phases should always be
in thermal equilibrium.
Consequently if a high degree of
thermal disequilibrium between phases does not exist, the
4-Eq model can give good results.
However in case of a
severe transient in which the vapor phase may exist in
subcooled liquid, the 4-Eq model may be a poor model to
use.
Since the code tends to fail in such a severe
transient due to numerical problems, the 4-Eq model gave
almost identical results to the 6-Eq model in the scope of
my experience.
The HEM model assumes large interfacial momentum
transfer so that the slip ratio is equal to one.
Practically
this is not a good assumption since the slip ratio is much
greater than one due to the high density ratio ( >2000 ) at
atmospheric pressure.
Thus the HEM model shows wide
discrepancr with low pressure experimental data.
THERMIT is a component code, which needs appropriate
boundary conditions at the top and bottom ends.
of boundary conditions can be used in THERMIT.
Two types
One is the
pressure-pressure set and the other is the pressure-velocity
set.
In case of loop simulation, however, the starting
point corresponds to the end point so that only one boundary
-----
--
-----
--
1111
I'W"11, I
-18-
condition is needed.
There are two experiments simulated in this thesis, an
ORNL natural convection experiment and a French forced
convection experiment.
The experiment reported in ORNL/TM-
-7018 is a single-channel natural convection sodium boiling
experiment that was performed in the Sodium Boiling Test
Facility of the Oak Ridge National Laboratory.
It is chosen
for simulation because of its simple geometry and well-documented output data.
The French experiment is a single
channel forced convection sodium boiling experiment performed
at the CEA, Grenoble, France.
The geometry of the French
experiment is simplified to have a uniform flow area and
calculation is done for the simplified geometry.
In Chapter 2 the details of the simulation methods are
given for these two experiments.
For the ORNL/TM-7018
experiment, first only the test section is simulated with
appropriate boundary conditions and subsequently the analysis
is extended to the whole loop.
In simulating the loop as a
whole, there are two methods possible, i.e. a simulated I-D
loop analysis and an actual 1-D loop analysis.
For the loop
analysis, condensation in the upper plenum should be taken into
consideration.
-sation occurs.
However the code presently fails when condenThe difficulty is circumvented by decreasing
the density ratio of the liquid and vapor phases artificially.
i
-19-
For this purpose, the vapor density is increased by 100 and
200 times with all the other properties intact.
These
changes in the density ratio correspond to actual pressures
The
of 10.0 Mpa (1470 psia) and 15.7 Mpa (2300 psia).
simulation of the French experiment is essentially the same
as the test section simulation of ORNL/TM-7018.
In Chapter 3, numerical and physical models are listed
and explained.
Physical models are the constitutive relations
that close the conservation equations.
4
As part of the numerical model, the energy loss problem
is considered.
With the present form of the energy conservation
equations, some portion of the total energy is lost due to
disregard for the frictional dissipation and some extra terms
that come from the mass transfer between phases.
As for the physical model, geometry plays an important
role.
The geometry of THERMIT is a rod-bundle geometry with
wire wraps, whereas the ORNL/TM-7018 experiment was done in
a single circular tube.
Hence either translation should be
done between the different geometries or constitutive rela-tions from the same geometry should be used.
In this thesis
the former approach is taken for all the calculations.
The
present contitutive relations and their problems are also
sumrmarized in Chapter 3.
In Chapters 4 and 5, all the computer results and
----I
-20-
experimental data are presented and analyzed.
For the test
section simulation, the difference of the pressure drops
between calculation and experiment is presented and the
results from the HEM, 4-Eq and 6-Eq models are compared and
discussed.
For the loop simulation an analysis explaining
why there should be an oscillatory behavior under two-phase
flow conditions is presented.
The oscillation period is
calculated analytically.
In Chapter 6, it is concluded that THERMIT is a good
numerical tool, although it still has some problems to be
solved.
Stabilization of the numerical scheme and verifica-
-tion of the physical models are major problems.
In Chapter 7, recommendations for future works are given
for further improvement of THERMIT and better understanding
of two-phase flow phenomena.
-21-
Chapter 2.
2.1
Simulation of the cases
Natural convection
Explanation of the experiment ORNL/TM-7018
2.1.1
This is a single-channel sodium boiling experiment
that was performed in the SBTF (Sodium Boiling Test Facility)
at Oak Ridge National Laboratory.
The object of the test
was to determine the maximum power that could be transferred
to the sodium coolant in an LMFBR fuel assembly subchannel
when the flow is driven by natural convection.
Fig 2.1 shows the whole test loop and Fig 2.2 shows the
test section.
The test section is composed of a heated zone
and an adiabatic zone.
Lower and upper plena are attached
to the ends of the test section.
The temperatures of the
lower and upper plena are fixed at 420 0 C and 590 0 C by sodium-to-air heat exchangers located inside the plena.
The upper
plenum pressure is always maintained at atmospheric pressure,
100 kPa
(14.5 psia).
As shown in Fig 2.1, the system is a loop with no bypass
flow.
The heat is input by a radiant furnace and measured
indirectly by measuring the furnace coolant temperature rise,
furnace coolant mass flow rate and the electric power input
to the furnace.
Since the heat input is the only controlled
variable, its accurate determination is important in analyzing
-22-
EQUALIZER
LINE
Fig 2.1
Sodium Boiling Test Facility - loop
ORNL/TM-7018)
(taken from
_
YY/
-23-
-TAYLOR
PRESSURE
TRANSDUCER (PT-7)
UPPER PLENUM TANK
:~Z,- FLOWMETER (FE-58)
VOLTAGE CONNECTION FOR
VOID DETECTOR SYSTEM
z
THERMOCOUPLES AND
VOID DETECTOR
0 E
VOLTAGE TAPS
4-.
U0
P<:,.U
/4"
Y
VOLTAGE CONNECTION FOR
VOID DETECTOR SYSTEM
LUC
Z
zC
O
0
t
wo
4:
w
z
PRESSURE
TRANSDUCER
(PE-2B)
(FE-B)
FOR
LOWER PLENUM
-TAYLOR PRESSURE
TRANSDUCER (PT-14AI
Fig 2.2
Sodium Boiling Test Facility - test section
(taken from ORNL/TM-7018)
-24-
the experiment.
It was found in the process of simulation
that there was a systematic error in measuring the heat input
to the test section.
2.1.2
Calibration of heat input
Table 2.1 shows the summary of the data given in the
report ORNL/TM-7018 for all the test runs performed in the
SBTF.
From Table 2.1 it can be observed that boiling began
when the test section power was about 900 W.
however, boiling began at a lower power.
In simulation,
Therefore the inlet
flow rate and heat input data in Table 2.1 are not consistent
because the energy convected out by sodium is not equal to
the heat input under steady-state condition.
There are two sources of error which may be responsible
for the deviation.
One is the error in the inlet flow deter-
-mination which might have underestimated the inlet flow rate.
The other is the error in the test section heat input which
might be lower than the values given in Table 2.1.
Permanent magnet flow meters are used to determine the
inlet flow rate.
The report ORNL/TM-7018 states that the
flow meters were calibrated after installation and the calcu-lated flow as determined by heat balance was within 15% of
the measured values in the forced flow tests.
Regarding the test section heat input, there is a high
''"-1111
1111111111
~
I
i
-25-
Table 2.1
SBTF natural convection results
(taken
Test
section
power a
(W)
Test
No.
from ORNL/TM-7018)
Heat_
flub
(W/cm2 )
Power
densityc
(I/cm 3)
Inlet
flow,
mean
(mt/s)
Outlet
flow
meand
(mt/s)
e
Frequencye
(Hz)
Phase
angle
(deg)
Single phase
107R2
109R1.
108R2
100R3
100R2
100R4
-300.
340
500
700
700
800
3
3
5
7
7
8
40
42
60
90
90
100
0.7
2.0
0.9
1.0
0.9
1.1
0.4
1.9
0.6
0.6
0.7
0.8
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
2.4
2.2
0.9
130
Chugging
131R1
1000
10
130
132R1
130R1
1100
1300
11
13
140
160
Steady boiling
129R1
1300
13
160
1.0
0.7
0.7
130
121R1
125R1
1400
1400
14
14
180
180
0.8
0.9
0.5
0.6
0.8
1..0
120
110
125R2
127R1
120R1
119RI
126R2
1400
1500
1500
1600
1600
14
15
15
16
16
180
190
190
200
200
0.7
0.8
2.4
1.1
0.9
0.4
0.6
2.1
0.9
0.7
1.0
1.0
0.8
0.5
1.2
110
110
130
140
100
122R1..
1600
16
200
128RI ,
1700
17
210
0.9
0.3
0.7
0.0
1.3
1.0
100
110
123R1 ,"
1700 '
17
210
1.0
0.4
0.7
0.1
1.1
0.9
100
110
124R1
1700
17
210
126R1
1700
17
210
Dryout
CFurnace power + furnace clamshell power - furnace coolant enthalpy rise - furnace
losses to ambient.
bBased on tube ID (98.6 cm2 ).
Ceased on heated coolant volume (8.0 cm 3).
dOutlet flowmeter was calibrated at inlet temperature (420*C) -not corrected for
actual temperature.
eCorresponds to dominant frequency based on power spectral density analysis of the
inlet flo~eter (FE-IB) signal.
-Angle by which inlet flow (FE-IB) leads pressure at inlet of heated section (PE-2B).
9Forced flow run used for flowmeter calibration check (see Appendix C).
lUnstable flow pattern; frequency is average of two distinct frequencies, 0.7 and
1.5 Hz.
iConditions preceding dryout.
Conditions during dryout.
-26-
probability of overestimation due to bending of the tube,
which might move the test section out of the focal plane of
the radiant heating source.
It is difficult to find out which error is responsible
for the inconsistency, but in this thesis it is assumed that
the heat input was overestimated.
The heat input as determi-
-ned by heat balance with inlet flow data is about 30% less
than the values listed in Table 2.1.
For all the code inputs,
70% of the listed values is therefore used as the test section
heat input.
The detailed procedure of the test section heat
input calibration is given in Appendix A.
2.1.3
Test section simulation
2.1.3.1
2.1.3.1.1
Steady-state
Boundary conditions
Initially only the test section is simulated with steady-state boundary conditions.
At the outlet, the pressure
boundary condition is applied because the pressure at the
upper plenum is always kept at atmospheric pressure.
At the
inlet, either the pressure or velocity boundary condition may
be applied using the values measured in the experiment.
After the steady-state has been reached, there is no differ-ence in the computational procedure between the pressure and
velocity boundary conditions at the inlet.
However in
IN,,
-27-
reaching the steady-state, the computation with the inlet
pressure boundary condition undergoes more severe transients
than that with the inlet velocity boundary condition.
For example a large reverse flow occurs at the inception
of boiling with the inlet pressure boundary condition, whereas
there is no reverse flow with the inlet velocity boundary
condition.
It is usually tougher to do the calculation with the p-p
boundary condition set than with the p-v boundary condition
set.
Thus the velocity boundary condition is chosen at the
inlet in spite of the fact that the inlet flow rate is not
well characterized.
As shown in the data of the experiment, every measured
parameter shows a highly oscillatory behavior and there does
not exist a steady-state in its strict sense.
However if
only a time averaged value is of concern, it is possible to
assume a quasi-steady-state with an average value of the inlet
velocity as the inlet boundary condition.
2.1.3.1.2
Geometry
The test section is composed of a heated zone and an
adiabatic zone.
Each zone is divided into five uniform meshes.
A mesh in the heated-zone is 0.194 m long and a mesh in the
adiabatic zone is 0.3 m long.
-28-
Since the geometry in the code is a rod-bundle geometry
which is quite different from the simple tube in the experi-ment, appropriate translation should be done between diffe-rent geometries.
In this thesis the constitutive relations for the rod-bundle geometry are used directly for the circular tube under
the assumption that the tube represents one fuel rod and the
coolant channel around the rod as in Fig 2.4.
The dotted line
in Fig 2.4 represents the symmetry line between the fuel rods,
and it is assumed that there is no interaction with adjacent
rods across the dotted line.
This assumption is true if there
is an infinite array of equally powered rods.
In the rod-bundle geometry, a pitch-to-diameter ratio and
a wire-wrap-lead-to-diameter ratio are necessary to calculate
the wall heat transfer and wall friction.
An equivalent pitch-
-to-diameter ratio in the circular tube geometry is calculated
in the following manner.
The coolant channel in Fig 2.5 can be made hydraulically
identical to the circular channel in Fig 2.6 under the follow-ing conditions.
(1) The flow areas should be same.
(2) The equivalent hydraulic diameters should be same.
-29-
adiabatic
zone
.
0.3 m
Ij
heated
zone
0.194 m
Fig 2.3 Geometry
for the test section simulation
-30-
P
i-
D -
Fig 2.5
Fig 2.4
Fig 2.6
Schematic diagram for geometry transformation
-31-
2
TrD
4
P
4
D
e
4
2
4
(p
(2.1)
)
=
D
De
(2.2)
Elimininating De from Eq 2.1 and Eq 2.2,
P/D = /1-2
= 1.2533
Since there is no wire wrap, a very large value is input to
the wire-wrap-lead-to-diameter ratio.
It may be better to use the correlations that are derived
from the same geometry, i.e. a circular tube.
However from the
practical point of view there is no need to use correlations
more directly applicable to circular tubes, because other
uncertainties seem to be greater than the uncertainty arising
from the difference in geometry.
2.1.3.2
Transient
The transient calculation in this section has the same
geometry and type of boundary condition as the steady-state
calculation in the previous section.
After the steady-state boiling has been reached, the
inlet velocity is oscillated with the period of one second.
-32-
One second is the typical period of the experimental data.
Fig 2.7 shows the variation of the inlet velocity in
the transient calculation and the experiment.
It should be mentioned that this is the transient for
the steady-state boiling, and not the transient between the
boiling inception and the steady-state boiling.
Whole loop simulation
2.1.4
The loop can generally be simulated in two ways, as a
simulated 1-D loop and as an actual 1-D loop.
The simulated
1-D loop is obtained by cutting some point of the loop and
extending it into a straight line.
The direction of gravity
should be determined according to the position in the loop.
The actual 1-D loop is obtained by putting a solid block in
the center of a two-dimensional rectangular pool.
2.1.4.1
Geometry and boundary conditions of the simulated
1-D loop
Fig's 2.8 and 2.9 show the geometry of the simulated
1-D loop analysis.
The upper face of the upper plenum in
Fig 2.8 is cut and the loop is straightened into a one
dimensional tube in Fig 2.9.
The boundary conditions at the top and bottom ends of
Fig 2.9 should be identical because they are the same point
-33-
Li)
cc
uU
LU
-j
_j
Z
LO;
a*,
LEGEND
o- PE-2B
I .
i
LL
-J
E
D
0
LEGEND
D- FE-SB
code
eS
E
a*a
0
*
..
*
,
,
input
LEGEND
0- fE-1B
i
I
I1
I
TIME (s)
Fig 2.7
Typical experimental data for steady-state boiling
experiment (taken from ORNL/TM-7018)
-34-
starting point of
simulated
Fig 2.8
Geometry for the simulated 1-D loop
1-D loop
-
---
-----
IEMI
-35-
fictitious cell
upper plenum
1
1
1
1
adiabatic
zone
T
test
1
section
1
1
1
heated
1
zone
1
I
1
lower plenum
1
return
-1
loop
-1
-1
-1
0
0
0
1
fictitious cell
Fig 2.9
Geometry for the code inputs of the simulated
1-D loop
-36-
in the loop.
Thus the pressure-pressure boundary condition
is used for Fig 2.9 so that the pressures at the top and
bottom boundaries are both atmospheric pressure.
Fig 2.9 shows that fictitious cells are defined at boun-daries according to the code format.
Since the boundary
pressures are defined at the center points of the fictitious
cells, not at the interface of meshes, an error enters the
calculation.
In order to reduce the error, very thin ficti-
-tious cells are used.
The length of the fictitious cells
is 1.0x10 - 1 0 m, whereas the length of the upper and lower
plena is 0.6175 m.
As can be seen in Fig 2.8, the direction of gravity is
different for different parts of the loop.
The direction of
gravity is defined at the interface of mesh cells and denoted
by the numbers ' 0, 1,-l' at each interface in Fig 2.9.
One of the major problems in simulating the loop experi-ment ORNL/TM-7018 was to keep the plenum temperatures cons-tant throughout each transient.
In THERMIT calculations the
plenum temperature cannot be determined a priori, but is
determined from the heat input and boundary conditions.
Since the temperatures in the upper
and lower plena are
kept at 5900C and 4200C in the experiment, a new scheme was
needed to keep plenum temperatures constant in the calculation.
- ------- ----
11 111
1
_IYI
IIi
- I
.
'
-37-
There are two ways of keeping the plenum conditions
constant.
One is to use the implicit scheme which solves
the conservation equations with the plenum temperatures at
the next time step already determined.
The heat extraction
from the plenum is determined as a result of the calculation
for each time step.
There will be no error of the plenum
temperature in this method, but a fundamental change of the
numerical scheme is required.
The other way is the trial and
error method which varies the heat extraction until satisfac-tory plenum temperatures are obtained.
Since the plenum
temperature decreases as the heat extraction increases, the
plenum temperature is a monotonously decreasing function with
respect to the heat extraction.
The unknown function between
the plenum temperature and the heat extraction is fitted by a
parabola and a new heat extraction is obtained from the
parabolic function.
The same procedure is repeated until the
required plenum temperature is obtained.
The details of the
method are given in Appendix C.
2.1.4.2
Artificial vapor density in the simulated 1-D loop
analysis
One of the major difficulties of the loop simulation is
that the code fails at the inception of boiling and when the
-38-
vapor reaches the upper plenum and begins condensation.
The
failure is a numerical problem induced by the high density
ratio, since the density ratio of the liquid and vapor phases
of sodium is over 2000 at atmospheric pressure.
In order to avoid the code failure, the density ratio is
reduced.by one hundred and two hundred times with all the other
properties intact.
Then the density ratios are approximately
24 and 12 and these correspond to the actual pressures of 10.0
MPa
(1470 psia) and 15.7 MPa (2300 psia).
With the increased
vapor densities the code did not have any numerical problem
and the results still showed the important characteristics of
the two-phase flow in a loop.
Note that the artificial vapor
density was used only for the loop simulations and not for the
test section simulations.
2.1.4.3
Geometry and boundary conditions of the actual 1-D
looD
The geometry of the actual 1-D loop analysis is shown
in Fig 2.10 and Fig 2.11.
It is a two-dimensional rectangu-
-lar array of cells with a coolant channel around an internal
solid block.
As Fig 2.8 and 2.10 show, the geometries of the
simulated and actual 1-D loops are exactly the same.
The coolant volume in any mesh cell should not be zero
due to the numerical scheme in the code, but the coolant
-39-
4 channels x 16 axial levels
top fictitious cells
upper plenum
adiabatic zone
heated zone
lower plenum
--/ ,;'/-
.
I A
Fig 2.10
I-, ,,' -
---------
bottom fictitious cells
Geometry for actual 1-D loop
-40-
open area
-II
II
-
' i
i
I
'
----
-
I---
IL,'
"
w-m
- upper
plenum
adiabatic
zone
internal solid block
I
---
heated
zone
lower
lenum
closed area
(
I--
II
I .
Fig 2.11
Geometry of
actual 1-D loop analysis
7e
u'li ulmi
I
nniullhlHlllllh
-41-
volume inside the solid block is of no relevance to the results.
Since the interfacial area with the solid block is closed,
there is no coolant flow across the solid interface.
The inter-
-facial area with the bottom fictitious cells is also closed
and the bottom boundary condition is the velocity boundary con-dition that the inlet velocity at the closed interface is zero.
However it does not matter what bottom boundary condition is
used, because there is no interaction between the bottom ficti-tious cells and the loop.
The interfacial area with the top fictitious cells is open
so that there can be a little vertical velocity component due
to thermal expansion of the sodium coolant.
The pressure at the
top fictitious cells is atmospheric pressure.
The upper and lower plenum temperatures are kept constant
using the same scheme as used in the simulated 1-D loop analy-sis, which is given in Appendix C.
Only a single-phase case has been run because the two-dimensional geometry of the actual 1-D loop takes much longer
computation time than the one-dimensional geometry of the
simulated 1-D loop.
However two-phase computations can be
made without any modification to this approach.
uiWn
-42-
Forced convection
2.2
2.2.1
Explanation of the experiment
This is a hypothetical experiment which is based on the
data of a forced convection experiment performed in France.
The French experiment could not be simulated with THERMIT
because of its complicated geometry as shown in Fig 2.12.
The momentum conservation equations in THERMIT assume that
the flow area is uniform in axial direction.
The variable
flow area in axial direction cannot be taken into considera-tion, unless the differential and difference forms of the
momentum conservation equations are altered.
Hence a hypothetical experiment with a uniform cross
sectional area, but with the same input parameters as the
French experiment was run with THERMIT to obtain calculational
results.
These calculational results cannot be checked with
the exoerimental data.
From the computational point of view, there is no diff-erence between the natural ,and forced convections except that
one
has a low-speed and low-power and the other has a high-
-speed and high-power.
2.2.2
Geometry and boundary conditions
Fig 2.13 shows the geometry of the hypothetical forced
convection experiment.
The simulation is done for two
-43-
D3 , MBP2 ,TIi1
D2,
Fig 2.12
French experiment facility
MSE
-44-
I
1x10 -6 m
fictitious cell
4 meshes in heated
zone and 6 meshes
in adiabatic zone
adiabatic
zone
0.183 m
N/
0.1Tm
heated
zone
1xl0
m
T
Fig 2.13
I
fictitious cell
Geometry of the hypothetical forced-convection
French experiment
IIll
-45-
different radii, r = 0.002 m and r = 0.003 m.
These are the
typical radii of the French experiment facility.
The test section is composed of two parts, a heated
zone and an adiabatic zone, and the heat input to the test
section is 10 kW.
The pressure-velocity boundary condition is used for the
top and bottom ends of the test section.
-46-
Chapter
3.1
3.
and physical
Numerical
models
Numerical models
3.1.1
THERMIT conservation equations
3.1.1.1
HEM, 4-Eq and 6-Eq models
THERMIT conservation equations for the two-fluid model
are:
Mass
ot (
-t[
Momentum
+ V- (p v ) =
vv
v
(3.1)
+ V-[(!-a)pv £] = -r
(1-()p
(3.2)
+ V.(ap v vv ) + CVP = -F
v vv
(a v)
dt (aPv v
SV
-F.
-aP
g (3.3)
(3.3)
+ (1-a) VP
Si[ (1-a)p v k] + V [(1-a)p v v ]
-F
Energy
w£ +F. -(1-)p
+ ". (oev
k
(3.4)
z
.3z
v
-ot (apvev v ) + V* (ap e v v v ) + PV *(av)
%v
+
v
+ a
+ P-
iCt
Q,, +Qi
(3. 5)
t [ (1-a)p e ] + V [(l1-a)p
-PC
= Qw
-Q i
£e£v
+ PV.[(l-a)v
(3.6)
IWOi
--- -- I-
-47-
The leftmost terms of the above equations represent
the rate of change of mass, momentum and energy in the control
volume when integrated over the control volume.
The P
term
in the energy conservation equation denotes the rate of work
done between the liquid and vapor phases.
These time derivatives are omitted in the following
analysis and only the steady-state with onewspatial variable
z is considered.
However the analysis can be easily extended
to the transient three-dimensional case.
Following equations represent the steady-state, one-dimensional conservation equations for the 6-Eq model.
Mass
-[(1-oa)p
Momentum
vz]
+
v2
Mz (aPv vz
-(ap
az
v
e
v v vz
(3.8)
= -
-3z
zvvz ] + (
-z (-a)p
Energy
(3.7)
-(ap vv vz ) =
Mz
-Fwv -F. i -ap~g
vgz
) + P- (v
vz
az
Pe vz
-F w
OzP
= Q
S[(1-a) -[(1-a)
(3.9)
+F i -(l-0)pg z (3.10)
(3.11)
(3.1
+Q
= Q
i
(3.12)
-48-
The previous set of conservation equations can be reduced
to four conservation equations with the assumption that the
liquid and vapor phases are always at a thermal equilibrium
state.
The constitutive relations that are related with the liquid
and vapor temperatures are:
- Ts)
s
(3.13)
c = C2 (T v - T )
(3.14)
ee
= C
1 (T
e
Qi = C 3 (T
-
c
(3.15)
Tv)
(3.16)
where
F
: interfacial mass transfer due to evaporation
I,c : interfacial mass transfer due to condensation
Qi : interfacial heat transfer
If it is assumed that the constants Cl , C2 and C 3 are very
large, the liquid and vapor temperatures are driven to the
saturation temperature, i.e. T
cannot cet infinite physically.
Ts
T=v T
, because r and Q
With the relation T
= Tv
determination of the interfacial mass and heat transfer
is not of interest, so that the phasic mass and energy
_~I',,
-49-
equations can each be added to yield mixture mass and energy
conservation equations.
This reduces to the set of conserva-
-tion equations to 4 from 6.
In the 6-Eq model, the parameters pv,p
or T
functions of T
T
and P,
, e v and e£ are
but with the assumption that
Ts, they all become functions of the saturation
= Tv
temperature Ts only.
This is shown in the following relations.
Pv = f(T
P
=
p
= f(TZ,P)
e
= f(T
e
= f(T2,P)
P
= f(T
s)
e
= f(T
s )
e
= f(T
e
= f(T
s)
T
= T
= T
f(Tv ,P)
,P)
P = f(T
Hence three unknowns T , T
one unknown T
s)
(3.17)
s )
s )
and P are reduced to only
and the number of conservation equations is
also reduced by two,
from six to four
yielding,
-50-
4-Eq model
(3.18)
8z [ap v v vz + (l-a)p RvzEz = 0
Ma s s
2P ) +
-z(apv vvz
Momentum
-z [1-a)p2
-z
z
-(1-a)
-F
wv -F. -ap vg
P
+
z
+ (1-a)
-F
(3.19)
wk +F. i
(3.20)
p gz
+ (l-a) p k e k v kz
-- [ap e v
v v vz
zZ
Energy
+Pz
9z
vz
vz
+ (l-a)vkz]
= Qw
+
Qwk
(3.21)
If the same procedure is repeated for the phasic momentum
equations, the HEM conservation equations result.
Here we
deal with the issue of mechanical equilibrium embodied in
the constitutive relation linking the phasic velocities.
F.
= C4 (vV
- v
)
(3.22)
where F.: interfacial momentum transfer
1
vz
Assuming C 4 goes to 4 infinity, the velocities v vz and
advz
should have the same value to keep F. bounded. This reduces
--
----
iii
-51-
the unknown velocities to a single homogeneous parameter
and suggests the combination of the phasic momentum equations.
Consequently the following set of HEM conservation equations
is obtained.
(3.23)
-[aPv
v + (1-a)p 2 ]v z = 0
az
Mass
2
aP
a
E[apv + (l-a)p z ]v z + -z
Momentum
-[ap
v
' (Fwv +Fw )
+ (l-a)p£)g z
(3.24)
+v
-ap e
Energy
where
+ (l-a)P
e]Vz + Pw
apv + (1-a)pk = Pm
aPvev + (l-a)p ek = emPm
3.1.1.2
Q
+Qw
(3.25)
: mixture density
mixture internal
: enerqy
Energy loss due to dissipation and interfacial
mass transfer
The suggested form of energy conservation equations
for THERMIT is:
-52-
9 (p
z
r
e v
) + P z (Cv)zvz = Qwv +Qi + f v
Pvevvz
+(F
Z
zg +
kkZ
wv
+F.)v
F
2
-zg z
z
v.z
(3.26)
z(l-)VRzQw
P
+zgzr +(Fwk
-Qi
F
2
2 VZz
- Fi)Vz
(3.27)
In order to satisfy the first law of thermodynamics, the
above form of energy conservation equations should be used.
The terms that contain F are produced by the transformation
of pressure drop into kinetic and potential energy increase.
If F were zero, Bernoulli's theorem would have resulted, but
with F not equal to zero, the additional terms with F should
appear.
The terms
r
2v
2
r 2
and f v
may be interpreted as kinetic
energy that mass should carry at the time of evaporation or
condensation.
However this is not the energy that is
interchanged between phases,.because the sum of the terms
7
2
(Vvz -v
2z )
is not equal to zero.
Rather it is the energy
produced by the transformation of momentum flux into kinetic
energy flux, due to the fact that there is a source term F
in the mass conservation eauations.
The potential energy terms with F are produced by the
transformation of gravitational force into potential energy
-
-~ ~IIIYllii
-53-
flux.
In the THERMIT energy conservation equations, frictional
dissipation terms were neglected because they were quantita-tively negligible.
atmospheric pressure,
However in sodium two-phase flow at
they cannot be neglected safely as
shown in the results of Chapter 4.
Pressure drop occurs due
to friction and that pressure drop is used in the energy
conservation equations of THERMIT.
Physically it means that
the thermal energy that is transformed from mechanical energy
due to frictional dissipation is lost to the environment.
Actually that portion of thermal energy is not lost, but
contributes to the increase of internal energy of the coolant.
These contributions are neglected in the THERMIT energy
equations.
Detailed derivation of the suggested form of
energy conservation equations is given in Appendix B.
Note that the old form of the energy conservation equations
has been used throughout this thesis.
3.1.2
of view
Explanation of the code from the numerical point
-54-
Overall solution scheme of THERMIT
6-Eq model for 1-D transient
Mass
-t
v ) =
v vz
~]
~[
(l-a)
)
+ --
v
t[+(l-a)p
Momentum
(p
(ap ) +
a
(ap
v
z
8
(aPvV
-F. -apv g
(3.28)
vtz
2
) +
(3.29)
= -r
-
P
-F
(3.30)
-55-
aa-[(1-a)p vz]+
a
=-Fwk +F i -(l-a)p
Qw
gz
(3.31)
+ P
Pza
e v v vz ) + P z (av)
(p
+
t (ap v eEnergyV
Energy
2 + (1-a)P3P
vz]
(l-a)
+Qi
t - (1-a)p e]
-Pt
+
(3.32)
-L[ (1-a) P e
z] + Pz
[
(l-a)vz
(3.33)
Qwk -Qi
The above momentum conservation equations have a
conservative form, whereas a nonconservative form is used in
the code.
In the nonconservative form, the a, pv and p£ are
outside the time derivatives and can be treated explicitly
utilizing the quantities at time step n, as shown in Eq 3.38.
The momentum is not strictly conserved according to the
difference form.of Eq 3.38, but the degree of nonconservation
must be tolerable in all the practical calculations.
The
nonconservative form of the momentum conservation equations
are,
av
Ov
atvz
+ an v v vz
vz +
az
P -F
az
wv
-F.
v
-ap
P
9z
(3.34)
-56-
- (1-a)
where
F.
= F. + Tv
Fi
=-F. - Tvz
iv
(P
(l-r)p£vQzZk z zt
£z
i a t
1
-F
(l-a)az
g
(3.35)
Vz
The difference forms of the one dimensional, six
conservation equations are set up as follows.
Mass
vapor
(aPv )
n+l
- ap
-[A(apv v ) v n
vz
liquid:
)
-
At
1
nn+
-{[A(ap
v ) v vz
V
a replaced by (1-a),
(3.36)
n+1/2
==
k-l/
k-l/
k+1/2
subscript v by k and
r by -F
Energy
vapor
(aove
v vv )
n+l
-
(ap e)
v v
n
(v
vz
) n+-
[pn
k+1/2
n+l ]
(vvz )
k-1/2
+
1
V
n[
+ -{Pn
At
(e
en
n
+(Pvevvvk+1/2
k /2 I[Aa
) k-/
v v k-1/2
n+l n
n a
At
[Aan
n+1/2
n+1/2
Qwv
+Q i
(3.37)
_
_ __~
-57-
liquid:
a replaced by (1-a), subscript v by £ and
Qi by -Qi
Note that the quantities without subscripts are at point k.
Momentum
vapor
+ l
[vnvz
n
(aPv) ki/2
v k+1/2
-(F
liquid:
- nv
Sz
z
n
k+/2
iv
n+1/2 -(ap
)n+/
k+1/2 -(v
]
vz k+1/2 + (ap ) n
n
k+ /2{(v
t
v k+1/2
vz k+1/2
n
k+1/2
) kn +
1/
2
k
zk+1/ 2
w
n+1/2
k+1/2
(3.38)
(3.38)
z
a replaced by (1-a) and subscript v by £
The momentum conservation equations are differenced about
the center of a face of a mesh cell whereas the mass and
energy conservation equations are differenced about the center
of a mesh cell volume.
In the above difference equations, some quantities are
required at positions where they are not defined.
These
quantities are calculated using the concept of donor-cell
differencing and spatial averages.
The specific approach
used is well explained in Ref.[1] section 2.2.2.
Now the difference equation forms are completely set
up and they should be solved numerically.
-58-
First the momentum equations are transformed to reduce
the velocities to spatial pressure differences.
In the process
the frictional terms (F
are treated
)n+1/2
iv k+l/ 2
and (F v)+1/2
wv k+1/2
semi-implicitly in the following way.
kn+1/2
1/2
iv )k+1/2
(Fi
=
[(v
) n+l
vz k+1/2
n+l
n+l
-
^
(Vz) K+1/2
iF iV
^
n+1/2
(F wv ) k+1/2
= [(v vz ) k+1/2 ] Fwv
where F.
and F
iv
(3.39)
(3.40)
are explicit quantities at time step n
wv
that should be given from constitutive relations.
The momentum conservation equations may be written as
follows.
n+l
n+l n+l
(Vvz) k+l/2arv + arv*tcv'delt + alpva-rdsdt (P+l-P kn+l
-arv*vv = -(v
deltw
) n+1
vz k+1/2
n+l
-(Vz) k+/2
de lt
n+l
(vz) k/
2
-arlvl1
-(v
.arl
]
-
(v
n+l
vz k+1/2
f iv
(3.41)
n+l n+l )
+ arl*tcl*delt + alpla.rdsdt. (P+l-
n+l
= -(v kz)k+1/2 .delt.fwl -
(v
n+l
)k+/2
n+1l
) n+
]deltfil
vz k+1/2
where the following notations have been used.
(3.42)
-59-
fiv
= F.
arv =
fwv = F
(ap )k+l/2
vv = (v
)k+/2
vz k+1/2
n
alpva =
k+1/2
rdsdt =
At
Ak+1/2
delt = At
The corresponding notations for liquid may be obtained
by replacing the vapor notation v with-the liquid notation £
and a
with (1-a).
In the Eq's 3.41 and 3.42 there are two unknowns
n+l
n+l
that can be expressed in terms
and (vzk+/2
(vz +l
vz kn+)
n+l
of the pressure difference (Pk+l
) at time step (n+l)
k+l
k
and other explicit terms at time step n.
Therefore the momentum conservation equations can be
linearized in the following form.
n+l
(vz
+l/2 = cpv
vz k+1/2
n+l
n+l
n+l
(Pk+1 - Pk ) + fv
n+l
n+l
(Vz)+
cp
(+l
- Pn ) + f£
hese
velocities are substitutedk+
k
(3.43)
(3.44)
These velocities are substituted in the mass and energy
-60-
conservation equations to get a set of equations with P, a, pv
Qw,
ek, F,
e
,
Qw
and Q.i at time step
(n+l).
Using the
equations of state and constitutive relations, the quantities
n+l en+1 ,
,
nn+l , e
n+l
S,
R
ev
v
n+1l
n+l and Q
Q
are approxiQ
n+l
w
n+l
F
wy
wk
-mated in terms of those quantities at time step n and the
first order derivatives with respect to the parameters a, T ,
T 5-and P.
pv
=
n+i
S
v
For example,
f(T V' ,
n +
= Pv +
Now
(3.45)
P)
Pv
aT
n+l
(T v
+
Sn
(P
T ) +
n+l
Pnn- P)
(3.46)
for the mass and energy conservation equations for
each phase, there are four unknowns a
n+l
n+l
Tv
T
n+l
n+l
and P
In matrix form, they may be written as follows.
C11
C12
C
13
014
C21
C22
C23
C2 4
AP
f
Acx
(3.47)
C4
C32
C3L
C34
C4
C4 3
C4 4
Eq 3.4
2
AT k
is a local matrix equation for one Newton
iteration step.
For each Newton iteration step, the coeffi-
-cient matrix on the lefthand side and the column vector on
- -
-
-- --YYI~
-61-
the righthand side should be recalculated.
matrix is inverted to eliminate Aa, 6T
The above local
and aT
in favor of
Then the global matrix equation over the entire
AP.
spatial mesh is set up and solved by the inner iteration
scheme.
The Newton and inner iteration convergence criteria
are both such that
AP between two successive iterates at any
spatial mesh point should be less than some preset values.
In case of a one dimensional problem, the inner iteration
is not necessary because the global matrix can be inverted
directly with little computation time.
Now the spatial pressure distribution at time step (n+l)
is completely calculated by the Newton.and inner iterations.
From the pressure distribution, the void fraction, vapor
temperature and liquid temperature can also be calculated
immediately and the calculation goes on to the next time step
(n+2).
Physical models
3.2
Physical models in THERMIT
3.2.1
3.2.1.1
Wall friction
For single phase, the THERMIT wall friction correlation
is
a mixture of Markley's laminar correlation[Ref.
Novendstern's
*
turbulent correlation[Ref.
4 ] with some
sodium version THERi.IT revised by Andrei L.
January,
1981.
4 1 with
Schor as of
-62-
appropriate transition formula.
The single-phase correlation
is extended to the two-phase case using the format of Autruffe's
None of the Markley's, Novendstern's and
correlation.
Autruffe's correlations is satisfactory by itself due to their
limited range of applicability.
Therefore these three corre-
-lations are combined to produce a new wall friction correlaThe present wall
-tion which has a proper applicable range.
friction correlation in THERMIT is given in the following.
M
1.034
0.124
-
(P/D)
6.94
Re
29.7(P/D) 2
2.239
0.086
0.885
(H/D)
P/D : pitch-to-diameter ratio
H/D : wire-wrap-lead-to-diameter ratio
Re : Reynold's number
M is a geometry factor and appropriate Reynold's number should
be used for the calculation of M for each phase.
The Reynold's
number for each phase is defined as follows.
(1-a)pP v De
t
Re
=
Re
ao v De
v v
(3.49)
(3.50)
vv
Since the wall friction is composed of two parts, i.e.
the liquid and vapor that contact the wall, the contact
fI
iYimiiliYIYnl
YIIilhiY
illlliiUllliWYIIi~1
iINIOiiiY
-63-
Before the dryout point,
fraction is defined for each phase.
only the liquid contacts the wall and the contact fraction for
vapor is zero.
1.0
'dry
(3.51)
S 1.0 - a
adry
a<C <1.0
1. 0-dry
(3.52)
cfv = 1.0 - cft
where adry = 0.957
With the above definitions the wall friction factor is expressed
in the following way.
For liquid
f32
P
D
lam, R
-trans,£
ff
Re
where
1.5 cfi
Re
turb,
-
2600
cfi
Re0.316M
0.25
turb,
/
: Re
Re
/1+ fl
lam,9
+
400
: 200,000
(3.54)
< 400
<Re
(3.53)
< 2600
(3.55)
400
? =
2200
For vapor
f
turb,v
0.316M cfv
Re0.25
Re
2600
5 Re
i 200,000
(3.56)
-64-
cfv
S32( P )1.5
/e
=f
f
Re
where
-
<400
(3.57)
f
400
lam,v
turb,v
trans,v
Re v
<Rev
< 2600
(3.513)
400
=
2200
Note that H is the wire-wrap-lead in [meter].
3.2.1.2
Interfacial mementum transfer
THERMIT has a Wallis-type correlation for the interfacial
momentum transfer which is given in the following.
4.31 P
2De
K
v -v
(l-)
[1+75(1-
)] } . 9 5
(3.59)
(3.60)
F . = K(vv - v)
where .De is the equivalent hydraulic diameter.
3.2.1.3
Wall heat transfer
The wall heat transfer is assumed to be composed of three
Darts, i.e. liquid convection, nucleate boiling and vapor
convection.
q = hvfc(twf-tv)
+ hlnb(twf-tsat) + hlfc(twf-t)
(3.61)
-65-
where
tv, t.,
tsat :vapor, liquid and saturation temperatures
hvfc :heat transfer coefficient through vapor convection
hlnb :heat transfer coefficient through nucleate boiling
hlfc :heat transfer coefficient through liquid convection
Three flow regimes are defined and the condition of existence
for each flow regime is as follows.
a) single-phase liquid
0.01> a
or
twf
tsat
b) two-phase annular flow
0.01
<a
<0.99
c) single-phase vapor
0.99
< a
The correlations used for each of these flow regimes are,
1) Schad's correlation for single-phase liquid
q = hlfc
Pe
(twf - t£)
k > 150
(3.62)
hlfc =
De rr
Pe £ < 150
hlfc - De
where rr is
defined in Eq 3.66.
(Pe
e
3
4.5 rr
(3.63)
(3.64)
2) Modified Chen's correlation for two-phase annular flow
q= hlnb
(twf -
tsat)
+ hlfc
Define the following parameters.
(twf -
t£)
(3.65)
-66-
rr
= -16.15
+
(P/D) [24.96
x 0.9 P
(-)
1-x
p
xtti
0.5
(-)
=
-
(P/D) 8.55]
v 0.1
(-)
(3.66)
(3.67)
tR > tsat
xtti
> 0. 1
xtti
<0.1
f = 2.35 (xtti + 0.213) 0.736
(3.68)
f
(3.69)
1.0
gx = (1-a)P IvL
t£
(3.:70)
< tsat
f = 1.0
gx = ap
(3.:71)
+
vv
(1-a)p~lvz1
(3.:72)
Definitions for the parameters sig, relx and retp.
sig = 9.8066 a
(3.73)
relx = gx De
(3. 74)
retp
= 1.0x10
< retp
<70.0
retp > 70.0
h
(f) 1 .25
s = 1/(1+0.12. retp
retp < 32.5
32.5
relx
= 0.0012 s
(3.75)
.14)
s = 1/(1+0.42 retp0.78 )
s = 0.1
k c
(ig
sig
(3.76)
(3. 77)
(3. 78)
)
0f .24
5 (Pre) -0.29 (p 0.25 ( _
)
( v
)
(3. 79)
v fg
-67-
Now with these definitions the heat transfer coefficients
are calculated in the following way.
hlfc =
kz
0.375
0.3
e- rr (f)
(Pe )
hlnb = h
s
h
_)0.75
sat
tsat v,
(twf - tsat) 0.99
(3.80)
(3.81)
3) Dittus-Boelter correlation for single-phase vapor
(3.82)
q = hvfc (twf - tv)
k
hvfc =
3.2.1.4
De
0.023(Pe ) 0.4(Re )0.4
v
v
(3.83)
Interfacial heat transfer
The interfacial heat transfer occurs by conduction and
convection'at the interface of the liquid and vapor phases.
Presently a very large constant is input for the heat transfer
coefficient 'hinter'.
qi
3.2.1.5
= hinter (tv - tk)
Interfacial mass transfer
The interfacial mass transfer model in THERMIT is a
modified form of the Nigmatulin model.
(3.84)
-68-
area;
True interfacial
a < 0.6
0.6
A
<a
4
-. 3 aM
=
t
A
< 0. 957
0.957
A
--
t
D
(3.86)
4[2/7(P/D*M) 2-aM0.5
D
=
t
4M
[2/7(P/DM)
2_M 0 5
-M]
1-a
0.5
(1-0.957
(3.87)
1
M =
where
(3.85)
D
2 / 7 (P/D)
2
-T
D : rod diameter
Note that
A = A t/3
Xe = 0.1
= 0.005
e
[
2
----3 Aa
c (1-.)/-R geX
/7
v fg
P
T -T
/
sT
T
/T
> T
V
s
(3.88)
0
3
c
r
2
2
ph
(1-)/-Rg
g c
T -T
v P fg
P
s
v
T
e
-
<
T
s
/T s
(3.89)
T
Finallyv
v
v
> T
s
(3.90)
~~___I__
_~__
Y
-69-
Problems of THERMIT physical models
3.2.2
3.2.2.1
Forced and natural convection
THERMIT physical models are derived from high-speed
forced convection experiments whereas ORNL/TM-7018 is a
low speed natural convection experiment.
It is not yet
known what amount of error will result, when.the forced
convection correlations are used for natural convection
cases.
For single-phase flow, one of the major differences
between the two cases is the velocity and temperature profiles
across the flow area.
The peak velocity in forced convection
is at the center region of the flow area, whereas the peak
velocity in some natural convection cases is somewhere between
the center and the wall.
This is because the coolant is heated
from the wall and the flow is driven by the buoyancy effect of
the heated'coolant.
When there are different velocity and
temperature profiles, different wall friction and wall heat
transfer correlations should theoretically be used.
For the two-phase flow, the situation is much more
complicated and it is difficult to say what the difference
between the forced and natural convection would be.
3.2.2.2
Flow regime
There are two flow regimes in the THERMIT correlations
for a sodium two-phase flow, i.e. bubbly and annular flow.
-70-
forced convection
Fig 3.1
natural convection
Typical velocity profiles for forced and natural
convection
-71-
Three points should be defined to determine the flow regimes,
inception of boiling, transition from bubbly to annular flow
and dryout.
In THERMIT physical models there are some inconsistencies
regarding the flow regimes as listed below.
1)
There is an inconsistency in determining the dryout
point.
For the wall friction, the dryout is assumed to occur
at
a = 0.957
as
1-aa
dry
and the contact fraction of liquid is defined
after the dryout point.
However for the wall
heat transfer the dryout occurs at a = 0.99 and the wall is
in contact only with vapor after the dryout point.
2)
There is an inconsistency in determining the transition
point between bubbly and annular flow.
The bubbly flow regime
is not defined for the wall friction and wall heat transfer,
but only fbr the interfacial area calculation.
The value
a = 0.6.is chosen for the transition from bubbly to annular
flow just to make the interfacial area a continuous function
with respect to the void fraction.
There is no physical
basis for the value a = 0.6.
3)
The droplets in the vapor core for an annular flow
should be considered.
The fraction of the liquid droplets
should be determined because the droplets significantly
affect the interfacial momentum transfer.
Since the
droplets may be assumed to flow at the same velocity as
-72-
the vapor, the interfacial momentum transfer is increased
in comparison with the flow without droplets.
4)
A physical model for the interfacial heat transfer is
Presently a very large value is input for the
required.
interfacial heat transfer coefficient, but it must be a
function of the interfacial area and flow regimes.
5)
There seems to be a problem in determining the interfa-
-cial momentum transfer.
In the present correlation, the
interfacial momentum transfer is proportional to the square
of the difference between the liquid and vapor average
velocities.
Actually the interfacial momentum transfer is a
local phenomenon at the interface and has nothing to do with
The difference of bulk average
the bulk average quantities.
velocities is sometimes a good measure of the velocity
gradient at the interface.
However if the liquid film
Reynold's number is above 3000[Ref. 6 1, the wavy motion of
liquid film is independent of the liquid film flow rate and
determined from the vapor ve-locity.
Hence the interfacial
momentum transfer may depend not only on the difference of the
licuid and vapor velocities but also on their absolute
magnitudes.
3.2.2.3
Condensation modeling
In THERMPIT, the liquid and vapor temperatures are
~----------
ii
-73-
artificially made almost equal by using a large interfacial
heat transfer coefficient.
According to the Nigmatulin model,
the interfacial mass transfer is proportional to the difference
between the liquid or vapor temperature and the saturation
temperature.
Consequently the interfacial mass transfer is
not reliable, because the liquid and vapor temperatures are
not reliable.
The advantage of the 4-Eq model is that physical models
for the interfacial mass and energy transfer are not necessary,
but the 4-Eq model cannot describe a state of thermal disequili-brium between phases.
Since a thermal disequilibrium state is
expected to occur in case of the condensation of superheated
vapor in subcooled liquid, the interfacial mass and energy
transfer is of primary concern for condensation modeling of the
6-Eq model.
Translation of physical models to different geometry
3.2.3
3.2.3.1
Equivalent hydraulic diameter
The wall friction and wall heat transfer are the transport
phenomena of momentum and energy from the wall.
The transport
phenomena from the wall are determined by the flow condition,
properties of coolant, wall surface condition and the geometry
of the wall.
It has been shown by experiments that in some
cases.the transport from the wall is relatively independent of
the geometry of. the wall, e.g. circular or rectangular,
-74-
provided the equivalent hydraulic diameters are the same.
If the transport phenomenon from one portion of the ---
perimeter affects the transport from another portion of the
perimeter, the equivalent hydraulic diameter cannot be used.
In other words, the equivalent hydraulic diameter cannot be
used when the flow area is too far from being circular or
the flow is laminar so that the wall effect is not uniform
over the flow area, but is a function of the distance from
the wall.
Consequently in translating the rod-bundle correlations
to the circular geometry, two conditions should be satisfied.
One is that the rod-bundle in LMFBR should not be too closely-packed so that the subchannel geometry is not too far from
being circular and the other is that the flow should be
turbulent.
Two equivalent hydraulic diameters are used in THERMIT,
the wetted equivalent diameter for the wall friction and the
heated equivalent diameter fer the wall heat transfer.
They
are defined as follows.
De(wetted) = 4A/(wetted perimeter)
(3.91)
De(heated) = 4A/(heated perimeter)
(3.92)
The equivalent hydraulic diameter may be a good approxi-mation if the previously mentioned two conditions are
-
-
~~II
mmII
I.II
1116
flil
W I it
-75-
satisfied, but the best way is to use the correlations that
are developed in the same geometry.
3.2.3.2
Interfacial area
The interfacial area in a rod-bundle geometry is given
in Section 3.2.1.5
'Interfacial mass transfer', and the basic
idea of the derivation is given in Ref. 4 .
If the same idea
is used for the circular tube geometry, the following
expressions for the interfacial area between liquid and vapor
phases are derived.
A
A
t
t
A
t
= 26
V
D
/
D
=4/(
.D
1-a
1 -adry
in bubbly flow
(3.93)
in annular flow
(3.94)
)0.5
(3.95)
after dryot
dry
where D is the diameter of the tube
In order to get a continuous interfacial area with
respect to the void fraction a, an approximation to the above
ecuations is made as follows.
At = - e- (.
D
t
At
4/
D
V-5
-
15
.
1.
(a
(adry
1.5
/---
1.5
(adry)
1
1.5
in bubbly and
(3.96)
annular flow
1-a
0.5
) (
(3.97)
) adry
after dryout
-76-
Fig 3.2 shows the difference between the interfacial
areas calculated by Eq 3.85, 3.86 and 3.87 with the equivalent
hydraulic diameter of the rod-bundle and by Eq 3.96 and 3.97
with the diameter of the tube.
Fig 3.2 shows that the inter-
-facial area between phases is dependent not only on the
equivalent hydraulic diameter but also on the actual shape of
coolant channel.
Although the equivalent hydraulic diameter may be a good
approximation for calculating the wall friction and wall heat
transfer, it is not so good for calculating the interfacial
area which determines all the transfer terms between liquid
and vapor phases.
3.2.4
Wall
Summary of physical models for circular tube geometry
friction
Autruffe's wall friction correlation[Ref. 5 ]
F
-0.2
0.18
2De (Re )(3.98)
F
0.2
v
(
Re
(3.98)
v
(3 99)
-0.2
(3.99)
V
2De
(-)
where
v
=
p
£De
Re
=
v
De
-77-
interfacial area
0.0
Fig 3.2
0.2
0.4
0.6
0.8
void
fraction
1.0
[Ea
Interfacial area for rod-bundle and circular tube
geometry with the same equivalent hydraulic diameter
-78-
FT =
Fk +
(l-e)F v
iadry
where (
=
6 =
-dry
Tadry
a > adry
Interfacial area between liquid and vapor
A
A
t
t
4/
D
(1.
4V(/1.5
D
5
/1.5
- 1.5
1
1.5)
(adry
/ 1.5 -
1
S 1.5
(a .)
1.5
in bubbly and
annular flow
1-a
-adry
after dryout
__
-79-
Table 3.1
Summary of the constitutive relations used
for the natural and forced convection simulations
Forced convec-
Natural convection
(ORNL/TM-7018)
constitutive
relations circular tube
-tion (French)
circular tube
relations
4-Eq
Modified Nig-
Interfacial
mass transf-
4-Eq
6-Eq
none
-matulin model
none
Eq 3.85 - 3.90
-er
Wallis correlation
Interfacial
Eq 3.59 -
momentum
3.60
transfer
large heat
Interfacial
energy
transfer
Axial wall
friction
none
transfer coeff.
Eq 3.84
Markley's, Novendstern's and Autruffe's
correlation
Eq 3.48 -
Wall heat
transfer
none
3.58
Schad's, Modified Chen's and Dittus-Boelter's
correlation
Eq 3.61 -
3.83
lNIlW,n
11,
i. ,,11l1i
ii lii llillm
lllilIH
I,
-80-
Chapter 4.
4.1
Results
Natural convection
The calculational results for the simulation of ORNL/TM-
-7018 are presented here.
There are two parts in the
simulation, the test section and the loop simulation.
For
the test section simulation, first the quasi-steady-state is
assumed and calculation is done for all test runs using the
average values of the oscillatory inlet velocity.
After the
quasi-steady-state (steady-state boiling) has been reached,
the inlet velocity is oscillated with the period of one
second and a transient calculation for the test run 129R1 is
performed.
For the loop simulation, the simulated and actual
1-D loop analysis results are presented.
In the simulated
1-D loop analysis, a hypothetical vapor density is used to
avoid the code failure.
Test section simulation
4.1.1
4.1.1.1
Steady-state
The steady-state here means the time average of the
oscillatory behavior as discussed in Chapter 2.
The pressure-velocity boundary condition is used for all
runs of the test section simulation.
The inlet velocity is
given from the inlet flow meter measurement.
-81-
In Table 4.1 the pressure drops in experiment and
calculation are given for all test runs.
The 4-Eq and 6-Eq
models gave almost identical -results and those results are
plotted in Fig 4.1.
Note that the experiment and calculation pressure drops
cannot be compared directly.
The calculated pressure drop
is between the center points of the fictitious cells,
whereas the experiment pressure drop is between the upper
and lower plena.
The condensation effect in the upper plenum
and the gravitational head between the center points of the
fictitious cells and the actual manometer locations are not
taken into consideration.
These effects are bounded and
discussed in Chapter 5.
In Table 4.2 the maximum number of Newton iterations,
the convergence criteria for Newton iterations and the
convergence criteria for steady-state are given.
The nega-
-tive number of maximum Newton iterations means that the
time step size is automatically reduced after the maximum
number of Newton iterations, if the Newton iteration conver-gence criterion is not satisfied.
If a positive number of
maximum Newton iterations is used, the calculation will
continue to the next time step, although the Newton itera-tion convergence criterion is not satisfied.
It may be
safer to use a negative number of maximum Newton iterations.
--
--
1111
-82-
Table 4.1
Steady-state test section pressure drop for
4-Eq/6-Eq models
Test
No.
Powera
[W)
inlet
AP calc
velocity
(4-Eq)
[cm/s ]
[BAR)
AP calc
(6-Eq)
APexp't
[BAR]
[BAR
107R2
210
8.438
0.2013
0.20
Single
108R2
350
10.849
0.1974
0.18
=hase
100R3
490
12.054
0.1930
0.08
100R4
560
12.260
0.1901
0.16
129R1
910
12.054
0.75149
0.75140
0.08
121R1
980
9.644
0.86342
0.86334
0.08
T wo
125R1
980
10.849
0.87640
0.87622
0.06
phase
125R2
980
8.438
0.82815
0.82829
0.08
127R1
1050
9.644
0.92575
0.92577
0.07
120R1b 1050
28.930
0.21965
119R1
1120
13.260
1.02340
1.02280
0.08
126R2
1120
10.849
1.02326
1.02315
0.08
128R1
1190
10.849
1.08605
Note
0.06
0.07
a 70% of the power given in the report ORNL/TM-7018
b In experiment it was a twb-phase run, but in calcula-tion it was a single-phase run with the given inlet
velocity.
-83-
Table 4.2
Steady-state test
section calculation convergence
criteria for 4-Eq model
a
epsn
b
delpr
c
Power
nitmax
107R2
210
-5
Single
108R2
350
-5
phase
100R3
490
-5
1.0x10-5
-5
1.0x10 5
-5
1.0x10 5
100R4
560
-5
1.0x10-
129R1
910
5
1.0x10-3
1.0x10-7
121R1
980
3
1.0x10-4
1.0x10-5
Test No.
5
-3
1.0x10-5
-5
1
l.Oxl0
1
-5
l.0x10
1.0x10-
5
-5
Two
125R1
980
3
1.0x10 - 3
phase
125R2
980
3
-3
1.0x103
1.0x10 - 5
-5
.O0x10
127R1
1050
3
1.0x10-4
1.0x10-5
120R1
1050
119R1
i ote
-3
-
3
1.0x10-3
1.0x10-5
1120
3
1.0x10-4
1.0x10-5
126R2
1120
3
128R1
1190
3
1.0x10-5
1.0x10-3
4 1190 3 -5
128R1
1.0x10
1.0x10
a maximum number of Newton iterations
b convergence criterion for Newton iteration
c convergence criterion for steady-state
(delpr = delro = delem)
-
Ii
-84-
[BAR]
P-P
lower plenum
Fig 4.1
upper plenum
Pressure distribution for the test run 129PR1
-85-
However it was impossible to satisfy any meaningful Newton
iteration convergence criterion at the inception of boiling
and condensation, whatever number of maximum Newton itera-tions might be used.
Although the Newton iteration convergence criterion is
not satisfied at the inception boiling, the calculation goes
on to the next time step and finally reaches the steady-state.
Hence the intermediate results are not reliable,
but the final steady-state does not depend on the interme-diate results, but only on the boundary conditions.
If the maximum fractional change in pressure, mixture
density and mixture internal energy between successive time
steps are less than delpr, delro and delem, it is concluded
that the steady-state has been reached.
In Table 4.3 it is shown how the heat input is trans-formed and partitioned among the enthalpy, kinetic energy
and potential energy rises of the coolant.
About 1"2% of
the total heat input is lost according to THERMIT calcula-tions for two-phase cases.
The reason has been given in
Chapter 3.
In Table 4.4 and 4.5, the liquid and vapor velocities
of the HEM and 4-Eq/6-Eq models are given.
W
-86-
Table 4 .3
Steady-state test section energy conservation for
4-Eq model
K.E. rise
P.E. rise
% energy
rise W]
[W]
[W]
loss
210
210
0.0
0.0
0.0
Single 108R2
350
350
0.0
0.0
0.0
phase
100R3
490
490
0.0
0.0
0.0
100R4
560
560
0.0
0.0
0.0
129R1
910
900
4.29x10-2
2.06x10-2
1.10
121R1
980
963
11.66x10-2
1.65x10-2
1.74
Two
125R1
980
965
8.79x10-2
1.85x10- 2
1.53
chase
125R2
980
962
15.11x10-2
1.44x10- 2
1.84
127R1
1050
1030
15.85x10 - 2
1.65x10 - 2
1.91
120R1
1050
1050
0.0
0.0
119R1
1120
1104
9.98x10-2
2.26x10- 2
1.43
126R2
1120
1098
" 16.56x10 2
2
1.96
128R1
1190
1165
21.67x10-2
1.85x10-2
2.10
.Test
Power
No.
[W]
107R2
enthalpy
1.85x10
0.0
-87-
Table
4. 4
model
Inter-
Steady-state HEM
for test run 129R1
velocity[m/s]
-face
Table 4.5
Steady-state 4-Eq/
6-Eq model for test run 129R1
slip
Inter-
ratio
-face
velocity[m/s]
slip
ratio
liquid
vapor
1
0.121
0.121
1.0
1.0
2
0.126
0.126
1.0
0.133
1.0
3
0.133
0.133
1.0
0.140
0.140
1.0
4
0.140
0.140
1.0
5
0.975
0.975
1.0
5
1.318
9.748
7.40
6
5.932
5.932
1.0
6
2.208
20.232
9.16
7
6.006
6.006
1.0
7
2.268
21.535
9.50
8
-6.977
6.977
1.0
8
2.327
23.372
10.04
9
8.402
8.402
1.0
9
2.400
25.621
10.68
10
10.738 10.738
1.0
10
2.464
28.483
11.56
11
15.394 15.394
1.0
11
2.605
32.163
12.35
No.
liquid
vapor
1
0.121
0.121
1.0
2
0.126
0.126
3
0.133
4
-
SUm
,.
-88-
Table 4.6
Dependence of the pressure
drop on interfacial friction
for test run 129R1
--
------- ---- - ~~.IIIYIYIIIIIII
1ili~~r
wll
WIYOIXlllkr
Mel
iIrllllbilWii~mrmlylIIWllii
-89-
4.1.1.2
Transient
After the steady-state has been reached, the inlet
velocity was oscillated with the period of one second.
Then
all the other parameters such as the pressure drop over the
test section and the void fraction, mixture enthalpy and
mixture'density at the end of the heated zone oscillate
according to the inlet velocity oscillation with some phase
shifts.
Fig 4.2 and 4.3 show the oscillations of those parame-ters.
It should be noted that the Newton iteration conver-
-gence criterion of 10-4 is satisfied at every time step in
this calculation.
Hence all the intermediate results are
reliable within the limit of the convergence criterion.
In Fig 4.4 the variation of the outlet liquid and vapor
velocities is shown during the transient.
4.1.2
Whole loop simulation
4.1.2.1
Simulated 1-D loop analysis
In order to avoid the failure of the code at the incep-tion of boiling and condensation, the vapor density has
been artificially increased by one hundred and two hundred
times as explained in Chapter 2.
The transient with a real
vapor density can be inferred from these two results.
Fig 4.5 and 4.6 are the simulated 1-D loop results of the
-90-
Vinlet
[m/s ]
0.24108
0.0
21
22
23
[sec)
P
[bar]
0.9
0.8
0.7
0.6
0.5
Fig 4.2
23
22
21
[sec]
Oscillation of the pressure drop according to the inlet
velocity
for test
run 129R1
-91-
Vinlet [m/s ]
0.24108
-
0.0
22
21
23
hImx10-3
[J/kg]
3
Pm [kg/m ]
.21
Fig 4.3
[sec)
22
23
[sec]
Oscillation of the void fraction, mixture enthalpy and
mixture density at the end of the heated zone for 129R1
-92-
inlet [m/s]
0.2
0.0
21
22
23
[sec]
v,out
vi,out [m/s]
3.5
2.5
1.5
0.5
22
Fig 4.4
23
Oscillation of liquid and vapor velocities at the
outIet
for the transient of the test run 129R1
[sec]
-93AP
[BAR]
Boiling
0.28 -
inception
0.27
vapor reaches upper
0.25
plenum
0.23
I
I
I
I
0
2
4
6
8
10
[sec]
10
30
50
70
90
110
I
AP
[BAR]
0.27
0.25
0.23
Fig 4.5
[Fig
4.5
Pressure
drop
over
the
test
secsetion
for
simulated
]
Pressure drop over the test section for simulated 1-D
loop analysis with vapor density x200
-94-
Vinlet
[m/s]
0.3
0.2
Boiling
0.1 -inception
0.0
-0.05
0
2
4
6
8
10
[sec
Vinlet
[m/s]
0.3
0.2
0.1
0.0
,
10
Fig 4.6
30
50
70
90
110 [sec]
Inlet velocity of the test section for simulated 1-D
loop analysis with vapor density x200
~~~~11,
-95AP
[BAR]
0.30
0.28
Boiling '
0.26
inception
0.24
vapor reaches upper
0.22
plerum
AP
[BAR]
0.26
[sec]
0.24
0.22
12
14
16
18
20
[sec]
Fig 4.7
Pressure drop over the test section for simulated 1-D
loop analysis with vapor density xl00
-96Vinlet
[m/s 3
0.3
0.2
0.1
Boiling
inception
0.0
-0.1
Vinle t 0
2
4
6
8
10
[sec]
0.3
0.2
-
0.1
10
12
14
16
18
20
[sec]
Fig 4.8
Inlet velocity of the test
section for simulated 1-D
loop analysis with vapor density xl00
_
IIYY
-97-
inlet pressure and velocity with the vapor density 200 times
increased.
Fig 4.7 and 4.8 are the results of the case
which has exactly the same geometry, boundary conditions and
inputs as Fig 4.5 and 4.6 except that the vapor density is
increased by 100 times.
In Fig 4.5 there is a peak of the pressure drop at the
inception boiling and the pressure drop decreases as boiling
goes on, because the hydrostatic pressure drop decreases
due to the lower density of vapor.
A minimum value of the
pressure drop is indicated when the vapor packet reaches the
upper plenum and condensation begins.
From that point on,
the pressure drop oscillates with the period of about 10
seconds.
amplitude.
Initially there is a damping of the oscillation
However after-about 70 seconds, no damping is
indicated-up to the point of 130 seconds.
The inlet velocity varies in the opposite way from the
pressure drop.
When the inlet velocity is minimum, the
pressure drop is maximum.
Consequently there is a phase
shift of 1800 between the pressure drop and inlet velocity.
Note also that the inlet velocity is minimum at the incep-tion of boiling.
Fig 4.7 and 4.8 show the same general tendency as
Fig 4.5 and 4.6, but much more severe transient at the
inception of boiling.
A reverse flow is indicated in
-98-
Fig 4.8 and the code failed at the time of 20 seconds.
In Fig 4.9 the steady-state boiling oscillation with
the vapor density 200 times increased is shown on a magni-fied scale.
The period is about 9 seconds.
Fig 4.10 through 4.19 show the spatial distributions of
the void fraction at discrete time intervals.
Fig 4.10
through 4.15 show the inception of boiling at the end of the
heated zone and propagation of the vapor packet and conden-sation in the upper plenum.
Meanwhile another vapor packet
is formed at the end of the heated zone and these two vapor
packets are connected and undergoes a similar vapor packet
propagation in the test section.
Fig 4.16 through Fig 4.19
show the void distribution in the steady-state boiling
region.
It can be observed that the shape of the void distr-
-ibution oscillates with the same period of 9 seconds as in
Fig 4.9.
4.1.2.2
Actual 1-D loop analysis
Fig 4.20 shows the steady-state pressure distributions
in the test section for the simulated 1-D loop and actual
1-D loop analysis.
It is the result of the single-phase run
107R2 the heat input of which is 210 W.
The steady-state inlet velocity is 0.070 m/s in the
simulated 1-D loop and 0.063 m/s in the actual 1-D loop
because the slope of the pressure distribution in the
-99-
[m/s]
[BAR]
0.2
0.25
period
%
1
_
0.24
100
9 sec
110
0.1
120
[sec]
Fig 4.9
Stable boiling oscillation in the simulated 1-D loop
with vapor density x200
-100-
0.6
3.0 sec
0.4
beginning of upper
0.2
plenum
0.0
heated zone
adiabatic zone
0.6
3.8 sec
0.4
0.2
0.0
heated zone
Fig 4.10
adiabatic zone
Spatial void distribution for simulated 1-D loop
with vapor density x200
-101-
0.6
5.0 sec
0.4
0.2
0.0
heated zone
adiabatic zone
0.6
6.5 sec
0.4
0.2
0.0
heated zone
Fig 4.11
adiabatic zone
Spatial void distribution for simulated 1-D loop
with vapor density x200
-102-
0.6
8.5 sec
0.4
0.2
0.0
heated zone
adiabatic zone
0.6
10.5 sec
0.4
0.2
0.0
heated zone
Fig 4.12
adiabatic zone
Spatial void distribution for simulated 1-D loop
with vapor density x200
-103-
0.4
13 sec
0.2
0.0
heated zone
adiabatic zone
0.4
14 sec
0.2
0.0
1
At
heated zone
A
m
adiabatic zone
0.4
16 sec
0.2
0.0
heated zone
Fig 4.13
adiabatic zone
Spatial void distribution for simulated 1-D loop
with vapor density x200
-104-
0.4
18 sec
0.2
0.0
heated zone
adiabatic zone
0.4
20 sec
0.2
0.0
heated zone
adiabatic zone
0.4
24 sec
0.2
0.0
heated zone
Fig 4.14
adiabatic zone
Spatial void distribution for simulated 1-D loop
with vapor density x200
ii i
1iIYII
III
-105-
0.4
28 sec
0.2
0.0
heated zone
adiabatic zone
0.4
32 sec
0.2
0.0
heated zone
Fig 4.15
adiabatic zone
Spatial void distribution for simulated 1-D loop
with vapor density x200
,
l
Ml
i lIUIIlliIIMIM
-106-
0.2
138 sec
0.1
0.0
heated zone
adiabatic zone
0.2
140 sec
0.1
0.0
heated zone
adiabatic zone
0.2
142 sec
0.1
0.0
heated zone
Fig 4.16
adiabatic zone
Spatial void distribution for simulated 1-D loop
with vapor density x200
I
_~_
_
I
_
____
i
-107-
0.2
144 sec
0.0
heated zone
adiabatic zone
0.2
146 sec
0.1
0.0
heated zone
adiabatic zone
0.2
148 sec
0.1
0.0
heated zone
Fig 4.17
adiabatic zone
Spatial void distribution for simulated l-D loop
with vapor density x200
-108-
0.2
150 sec
0.1
0.0
heated zone
adiabatic zone
0.2
152 sec
0.1
0.0
heated zone
Fig 4.18
Spatial
adiabatic zone
void distribution for simulated 1-D loop
with vapor density x200
_
~~_ ~ ~
-109-
0.2
154 sec
0.1
0.0
heated zone
adiabatic zone
0.2
156 sec
0.1
0.0
heated zone
Fig 4.19
adiabatic zone
Spatial void distribution for simulated 1-D loop
with vapor density x200
-110-
pressure
upper plenum
Fig 4.20
lower plenum
Pressure distribution for single phase loop analysis
for the test run 107R2
-
'Y
"'-
-111-
simulated 1-D loop is a little steeper
as shown in Fig 4.20.
Since the simulated and actual 1-D loop have the same
geometry and heat input, they should give exactly the same
result.
The deviation in the steady-state pressure distri-
-bution and inlet velocity of the two analyses is due to the
different treatment of the transverse wall friction.
In the
simulated 1-D loop, all the wall friction is treated axially,
whereas in the actual 1-D loop
a separate correlation is
used for the transverse wall friction.
Another source of the deviation in the velocity is the
thermal expansion of sodium.
In the simulated 1-D loop it
adds to the axial velocity, whereas in the actual 1-D loop
it adds to the vertical velocity component at the top face
of the rectangular pool.
However the thermal expansion
effect has no relevance for the steady-state results.
4.2
Forced convection
The calculation results for the simulation of the
French forced-convection experiment are presented here.
Simulation is done for the hypothetical geometry of uniform
flow area in axial direction.
Table 4.7 and 4.8 show the calculation results for two
-.
different radii tubes, r = 0.002 m and r = 0.003 m.
Fig 4.21 is the plot of the results on coordinates of inlet
mass flow rate and total pressure drop over the test section.
Table 4.7
Run
Forced-convection calculations for r = 0.002 m
Vinlet
(m/s]
m
AP[BAR]
(kg/s]
enthalpy
rise[kW]
max.
slip
max.
void
ratio
fraction
% loss
in energy
1-2
10.0
0.106
3.12251
9.962
1.0
0.0
0.38
2-2
5.0
0.053
1.C2615
9.994
1.0
0.0
0.06
3-2
4.0
0.042
0.73875
9.996
1.0
0.0
0.04
4-2
3.0
0.032
0.49989
9.998
0.0
0.02
5-2
2.0
0.021
0. 31207
9.999
1.0
0.0
0.01
6-2
1.75
0.019
0.27329
9.999
1.0
0.0
0.01
7-2
1.5
0.016
0.67066
9.996
3.01
0.8192
0.04
8-2
1.4
0.015
1.13522
9.985
3.64
0.8690
0.15
9-2
1.3
0.014
1.72212
9.969
4.18
0.8916
0.31
10-2
1.2
F A I L
-112-
'1.0
Table 4.8
Run
Forced-convection calculations for r = 0.003 m
vinlet
[m/s]
AP[BAR]
[kg/s]
enthalpy
max. slip
max. void
% loss
in energy
rise[kW
ratio
fraction
1-3
4.44
0.106
0.57404
9.993
1.0
0.0
0.07
2-3
2.22
0.053
0.26603
9.998
1.0
0.0
0.02
3-3
1.78
0.042 "
0.22303
.9.999
1.0
0.0
0.01
4-3
1.33
0.032
0.18641
9.999
1.0
0.0
0.01
5-3
0.89
0.021
0.15550
10.000
1.0
0.0
0.0
6-3
0.78
0.019
0.14835
10,.000
1.0
0.0
0.0
7-3
0.67
0.016
0.39279
9.992
3.83
0.8603
0.08
8-3
0.62
0.015
0.60586
9.978
4.76
0.8971
0.22
9-3
0.58
0.014
0.81121
9.957
5.60
0.9173
0.43
10-3
0.53
0.013
1.00211
9.930
6.42
0.9308
0.70
-113-
-114-
power = 10 kW
heated zone = 4 meshes (0.15 m each:)
adiabatic zone = 6 mreshes
AP
(0.183 m each)
[BAR]
4.0
3.0
2.0
1.0
r = 0.002 m
f-
.
r = 0.003 m
___
0.0
0.0
0.02
0.04
0.06
0.08
___
0.1
[kg/s
mass flow rate
Fig 4.21
S-curve for two different flow areas for the
hypothetical French experiment
---
---
~
,ii
IIIYIIYYYIIIIIII
lul
-115-
Note that the minimum point in the so-called S-curve is the
boiling inception point.
The code failed at some.point in the two-phase region
and the inlet mass flow rate could not be reduced any more.
As the physical models and the numerical scheme are improved
to the state that the S-curve can be completed to the zero
mass flow rate, the suggested form of the energy conservation
equations in Chapter 3 would have to be used.
A higher velocity will result in a higher frictional
pressure drop and more dissipation of the mechanical energy
into thermal energy.
Hence the fraction of the dissipation
thermal energy will increase.
At high inlet velocities the flow is single-phase.
As
the inlet velocity is decreased and the flow remains single-phase, the energy loss decreases.
However upon further
flow decrease, boiling begins and the vapor slip increase
offsets the flow decrease effect and
the net energy loss increases.
The results in Table 4:7 and 4.8 are an indication that
the energy loss problem of the present energy conservation
equations will become more severe as the calculation goes to
a higher void fraction region.
--
I__
~ml.
-116-
Chapter 5.
Discussion of the results
Natural convection
5.1
Test section simulation
5.1.1
5.1.1.1
Steady-state
5.1.1.1.1
Pressure drop
The only parameter that can be checked against experiment
in the steady-state is the pressure drop over the test section.
Fig 4.1 shows that there is a considerable deviation between
the 4-Eq/6-Eq model results and experimental data.
The HEM
model is even much worse than the-4-Eq/6-Eq model.
The calculated pressure drop is determined by the physical
models in the code and the specific way that the experiment is
simulated.
The physical models directly related with the
pressure drop are the wall friction, interfacial mass transfer
and interfacial momentum transfer
There may be two sources of error in the pressure drop
calculation.
One is the error in the physical models related
with the pressure drop and the other is the error in simulation
of the experiment.
In simulating the test section, some features of the
experiment are not taken into account.
Since the sodium vapor
condenses completely in the upper plenum, there is a pressure
rise due to deceleration of sodium.
The upper plenum in the
-117-
experiment is a single-phase liquid region, whereas the top
end in the calculation is a two-phase region.
Thus the
oressure rise due to deceleration should be considered as an
additional term.
Another source of error in simulation is whether the
exact positions of manometers coincide with the mesh centers
If the manometer position
where the pressure is calculated.
does not coincide with the center point of a mesh, the
pressure difference due to gravity should be taken into
consideration.
These two errors are bounded for the test run 129R1.
(position deviation)
<
0.5 m
(pressure drop due
to position deviation
<
+
(pressure rise due) = m[xv
to deceleration
4000 Pa
(l-x)v ]/A
v +
= 540 Pa
<
1000 Pa
Therefore
Total deviation in pressure
drop due to simulation error
<
5000 Pa
(0.05 BAR)
The simulation error estimated as 0.05 BAR does not
account for the deviation in Fig 4.1.
Hence it may be
concluded that there is a problem in applying the physical
1111
-118-
models in THERMIT to the case of ORNL/TM-7018.
First it is questionable whether the high-speed forced
convection correlations in THERMIT give a good result for the
low-speed natural convection experiment of ORNL/TM-7018.
It is
also questionable whether the rod-bundle geometry correlations
will be applicable to the circular tube geometry.
One or more of the physical models may be inappropriate
due to the different flow condition or geometry, but it is
impossible to know which physical model is responsible for the
For validation of the physical models,
deviation in Fig 4.1.
more experimental data are necessary to check with calculation.
The sensitivity of the pressure drop to the wall friction
and interfacial momentum transfer will be discussed here.
Some intuition for the two-phase wall friction pressure drop
can be derived from considering the single-phase case.
The
single-phase expression for the wall friction pressure drop is,
2
pv
AP = f L
De 2
(5.1)
The wall friction factor f is vital for accurate pressure drop
prediction, but the calculation in Appendix D shows that f is
rather insensitive to the flow condition according to THERMIT
wall friction correlations.
The error in the wall friction
factor is linearly propagated to the pressure drop, i.e. 1%
error in f results in 1% error in the pressure drop.
-119-
As for the interfacial momentum transfer, it does not
affect the pressure drop directly before dryout, but does
affect it through the liquid velocity.
If there is more inter-
-facial momentum transfer, a higher liquid velocity will result
A higher liquid
due to the increased friction force on liquid.
velocity will again give rise to a higher pressure drop as
Eq 5.1 shows.
The interfacial momentum transfer is an import-
-ant parameter in determining the two-phase condition.
It
establishes the flow quality and the void fraction, and all the
other physical models are strongly dependent on that void
fraction.
Moreover the transferred mass carries its own momentum so
that the interfacial momentum transfer is composed of two parts,
friction forces and momentum carried by the transferred mass.
Sensitivity of the pressure drop to the interfacial
momentum transfer is given in Table 4.6.
Very large interfa-
-cial momentum transfer will result in equal velocities of
liquid and vapor, therefore the HEM model.
The code tends to
fail as the interfacial momentum transfer decreases.
5.1.1.1.2
Energy conservation
Table 4.3, 4.7 and 4.8 show that a certain portion of the
total energy is lost in THERMIT calculations.
The reason for
the loss has been given in detail in Chapter 3.
-
----- I
-120-
The fraction of the lost energy increases as the total
heat input increases.
This occurs because at a higher power
there is a more interfacial mass transfer and more frictional
dissipation due to increased velocities.
The importance of the energy conservation can be
illustrated in the following way.
A 1% error in the energy
corresponds to 1% error in the flow quality.
That 1% error
in the flow quality may cause a considerable change in the
flow condition due to high density ratio between the liquid
and vapor phases of sodium at atmospheric pressure.
Especially
for the dryout point determination, a small amount of error in
energy may significantly affect the prediction.
5.1.1.1.3
Comparison of HEM, 4-Eq and 6-Eq models
Table 4.4 and 4.5 show the velocities of the liquid and
vapor phases for each model.
The HEM model is too far from
experimental data, therefore completely erroneous.
As can be seen all through the results, the 4-Eq and
6-cE
models are almost identical.
The 6-Eq model can be
reduced to the 4-Eq model with the assumption that the liquid
and vapor temperatures are equal to the saturation temperature
in two-phase region as explained in Chapter 3.
Presently
THERMIT uses a very large constant for the interfacial heat
transfer coefficient so that the liquid and vapor temperatures
-121-
are made almost equal to a saturation temperature.
Under such
a condition, there cannot be much difference between the 4-Eq
and 6-Eq model results.
5.1.1.2
Transient
Fig 4.2, 4.3 and 4.4 show that all the parameters
oscillate with the same period as the inlet velocity, although
with some phase shifts.
The phase shift is a characteristic
of each parameter and is the final result of all the interac-tions between phases and between the coolant and the wall.
Consequently there is no reason that the phase shift of one
parameter should be equal to the phase shift of another.
Fig 4.2 shows that the maximum point of the pressure
drop corresponds to the minimum point of the inlet velocity,
which means a 1800 phase shift.
Loop analysis results in
Chapter 4 show the same behavior between the
P and v.
Inlet"
In Fig 4.21, the slope of the S-curve is negative in the
two-phase region.
If the inlet mass flow rate increases, the
pressure drop should decrease and vice versa.
Hence the
pressure drop should be maximum when the inlet velocity is
minimum as Fig 4.2 shows.
SWOIIIUI
-122-
5.1.2
Whole loop simulation
5.1.2.1
Oscillation
The simulated 1-D loop analysis has shown that the
outputs of the system are oscillatory although all the inputs
to the system are constant with respect to time.
The experi-
-mental-data also show that the measured parameters oscillate
with a certain period, although the inputs are constant.
For the test section simulation in section 5.1.1, the
output is constant with constant boundary conditions.
For the simulated 1-D loop analysis, however, a small
perturbation from equilibrium state does not damp exponentia-11ly,
but continues oscillating with constant amplitude.
The
oscillatory behavior of the two-phase flow in a loop experi-ment is an inherent characteristic of the system.
In early 1960's many people [Ref. 11 - 14] tried to make
a general explanation of the two-phase flow oscillations, but
were not successful due to lack of knowledge about two-phase
flow phenomena and too many parameters that make the analyti-cal analysis impossible.
The calculation of THERMIT with the vapor density 200
times increased has shown that the system keeps on oscilla-ting with a certain period and amplitude after the steady-state boiling has been reached.
-123-
Mathematical model of oscillation
5.1.2.2
A mathematical model of
in
the two-phase
a loop is suggested in this section.
flow oscillation
idea of the
The
approach is to divide the loop into two regions, the test
and the return loop, and set up mass, momentum and
section
energy
conservation equations
for both regions.
The test
are
section and the return loop conservation equations
coupled through an empirical relation which is obtained from
shown in Fig 5.3.
the calculation results and
Conservation equations integrated over the test section
Mass
Ms[VvPv +V p
+V
-
P]
+ a oPvov A + (1-a 0)P
u
= 0
P.iU iA
0A
(5.3)
Energy
E -[VvP
Ot
e
v v v
+
+V p u
(1-a 0)
+V
p se
s
Si
A -
p iu
s
uh
] + a p
v0
0
ih i
A
u
h
vO v0
A
= Q
(5.4)
Momentum
3 [VpP
v v v
+V £ P u £z +Vsk s
2
u
s]p]
+
2
0 P v0 uvv
A
2
A -
iu 2.A + (P0-Pi)A = -F
+ (l-
)P
-[VvPv
+Vp% +Vs£Ps]g
where Qw is the total heat input and Fw is the total
wall friction force in
upvard direction.
(5.5)
-124-
hvo, Uvo
PvO'
h
0'
Uo '
P
O
T
two-phase
region
single-phase
region
i
hki'
Fig 5.1
uzi'
@i
Schematic diagram of the loop and the test section
-125-
Notations
h0
h
v0'
0
uv0' u0
enthalpy, density, and velocity
of each phase at the outlet
h
, Pi'
i : enthalpy, density and velocity of liquid
at the inlet
a0 : void fraction at the outlet
A : cross-sectional flow area
Vv, V
,
V s k : volumes of vapor and liquid in the two-phase
region and the liquid in the single-phase region
ev, eZ,
Pv' Pi' Uv, u
: spatial averages of internal energy,
density and velocity of each phase in the two-phase region
esk' Ps'
Usz : spatial averages of internal energy, density
and velocity of liquid in the single-phase region
PO : pressure at the bottom face of the upper plenum
PO : pressure at the top face of the upper plenum
PI
: pressure at the bottom face of the lower plenum
: pressure at the top face of the lower plenum
P.
1
_
~
IL
-- -- ...
-126-
Conservation equations at equilibrium state
0
at =
0 Pv
Mass
Energy
Uv0 A + (1-a 0 )P 0 u 0 A - P iUiA = 0
aP v0uv0hvOA + (1-a 0 )P
0u
0 h£ 0 A
- Piu
(5.6)
h
A
Qw
Moemntum
ap
0
(5.7)
-2
-2
u
u2 A + (1-a )p
vOvO
t0
0
= -F
-
[Vv
A
0
-
-2
p iU i
A
k
-
+ (P'-fP)A
+
0 1
+V p% +V sP 5 s]g
(5.8)
In the above equilibrium state equations, it is assumed
that UvO , uk0 , u i, Vv, V£, Pi and Fw are time-dependent
variables and all the other parameters are constant.
When the equilibrium conservation equations are subtracted
from the time-dependent conservation equations, the perturba-tion equations are obtained.
Perturbation equations
ass
1
p
v2t
+
-(V-VZ)
(V -V ) + p
£7(VEVR
v v
(1-a
0
)P
0 (u t 0 -u
0
)A
+ apv0(u v0-uv )A
a 0 P 0 vO v0
-
P i(u
i-u i)A = 0
(5.9)
-127-
Energy
+
P
-
(u ouo)h0A
a0)
(1-)P0
+
(V -V)
-V ) + p et
ve -t(V
0
(u
v 0( u v 0 - u v
-u
)h
o) hv
0A
A = 0
(5.10)
Momentum
p uv-t(V V-v
+ p
t(V
)
+ Ps
+pV f-(u-u
-p
2
(u
2
-2
)A -
-u
(P.-P)A
[(V
-V
v
V
)p
)p
(V -V
+
v
-2
0 (uO-£0)A
= -(F w -F
1i
i
1i
i
Since V
s-Vst (us-Us
-2
2
(U -UV )
+ P VV-
)
vO (uv 0 -u v0)A + (1-P0
+
-
-V
(5.11)
]g
is assumed to be constant, V
v
+ Vz =
(constant).
Thus
Vv -
5t
(v
k
-
= -(V
V
(v
-
9
=v
)
(5.12)
V )
v
- -t v -
(5.13)
Vv )
The certurbation equations may be written as follows.
Mass
(V
(--
)~t
+
(1-a0) P
-v
(u
)
uO
v(Uvo-v)A
vO
+
0 -u
0
)A
-
p
i(u
i-u
i)A = 0
(5.14)
__
__
I
I[!
I_
I II
Ii
-128-
(Pvev Pe )-FRSt
v v
Energy
(1-a) P
0
Momentum
0
t0
(U
aP 0(u -u )hv0A
0 vO vO vO vO
0 -u 0 )h
k0
tO
v
v
to
A - P
t
v
k
(u-u)
v-
Rii
)h
+pV
v
k
-2
) +(2
(u 0-Uv
+ aOp
P
A
=
0
(u -u )
ZP~VRE
(Pi-Pi)A
z)g
v )(pv-
R
)A
2 -2
p i(u i-u i)A -
2 -2
+(1-a 0)P0(u 0-u 0)A -
= -(Fw-Fw ) - (V
(u .-u
zi
(5.15)
) + pV v
(U
Vs
+
v
v
(v-v
v-pu)
(Pu
v v-PkRR
+ P
)
(V -v
(5.16)
The momentum conservation equation is simplified to make it
a linear differential equation.
(P
vv-
-p
+ P
(u
Vs
+ 2(1-
-
+
V-V)
k k) -tVv v
(P -P
0
-Us
£0(u
) PRUo
)A
= -(F
PVv
v v5t
) +
2a
0
0 -u£0)A
-F
)
-
(u-u
v
PUv
-
2p
(V -V
kV
v)
+
(u
0-u
iU
i(u i-u i)A
)(pv-p
-(u
-
-u)
)A
)g
(5.17)
Conservation equations integrated over the return loop
For the return loop, the mass and energy conservation
eauations are not needed because the return loop is adiabatic
-129-
p0
1
Fig 5.2
Return loop
will
-130-
and has uniform flow area.
Mass
u
Energy
er is constant all through the return loop.
Momentum
is constant all through the return loop.
+
2 (Mu)
F-t
rr
(P!-P )A = Mrg g-
1
0
(5.18)
F rw
Momentum at equilibrium
(P -P
i
0
)A = -M
rg
g -Fr
(5.19)
rw
Perturbation equation for momentum
-u)
Mr(U
+
(5.20)
(Pi-Pi)A = -(F rw-Frw)
Assume that
ur = Cuti
(5.21)
Since the temperatures of the coolant in the return loop
and lower plenum are constant with respect to time, C is
constant.
Note that C is greater than one because the sodium
The following
density increases as the temperature decreases.
momentum conservation equation for the return loop is obtained.
M
(u
rC-atuii
-
i
) +
(P.-P )A
=
(F -rw-
rw
(5.22)
-131-
Now the conservation equations for the test section and
the return loop are completely set up and they should be
combined together to produce a single ordinary differential
equation with respect to time.
In the process, the following
additional assumptions are made.
v = Uv
u
= U0
(5.23)
us
= ui
.
P.P.
1
1
- P
= Pf
1
1
Up to this point, four Eq's 5.14, 5.15, 5.17 and 5.22 with
P.-P. and V -V are
five unknowns u -UvO u
v0
obtained.
0-u u i-
0
v0'
1
1
and V-V
are
Hence we need an additional relation between the
unknowns and it is provided by the empirical relation between
and VV -V V in Fig 5.3.
P.-P.
1
1
Fig 5.3 is
a plot of the THERMIT
calculational results and is expressed by the following Eq 5.24.
P.
1
-
.
1
A
g(p
-p- v )(V v -V v )
g
(5.24)
where K = 0.5
Now the set of Eq's 5.14, 5.15, 5.17, 5.22 and 5.24 can
be solved with following
wall friction correlations.
-----
--
--
liiii
-132-
(u .-u i )
Frw - F rw = f (U11
F
- Fw
=
f 2(u i-ui
)
+
(5.25)
f3 (u -u
)
(5.26)
After some manipulations, the following second-order ordinary
differential equation with respect to V v -Vv results.
2
dt
(Vvv-
)
+ C
v v + C2v
C2
2 (V-Vv ) = 0
1 d (V-V)
(5.27)
where C 1 and C2 are constants.
Eq 5.27 yields a simple oscillation if the following condition
is satisfied.
2 << 4C
1
2
(5.28)
Then the period of the oscillation is,
T
2
2
(5.29)
-133-
------ - (V -9
AP [Pa]
(Pi-Pi
)
g (P-Pv ) /A
)
100
r
r
I
I '
I
\
r'
I
\
I
I
1
I
I
If
I
\
I
r\
I
50
n:
0.0 -
-50
I
I
t
I
I
I
I
I
I
t
I
I
1I
I
I
S
I
I
1
I
I
I
1
I
I
1
II
I
.9
.9
I
\
I
t
r
.I
II
I
.
r
rI
r
r
I
!/
.9,
\
\
\.
-100
Fig 5.3
I
L
I
..I
110
120
130
140
Calculation results that show
and V -V
between P.-P.
V
V
1
1
150
[sec]
the relationship
Ii,
-
-134-
Chapter 6.
Conclusions
THERMIT is a good numerical tool for predicting and
analyzing sodium two-phase flow phenomena, although there
are some problems to be solved.
The two fluid model of THERMIT is a versatile model that
can describe a wide range of flow conditions.
However, the
constitutive relations of the two fluid model are not in a
well-established state, especially the interfacial mass and
energy transfer terms.
To circumvent the difficulty, the 4-Eq model is deve-loped, in which the interfacial mass and energy transfer
occur instantaneously to maintain thermal equilibrium betw-een phases.
The numerical scheme of THERMIT works well for a low
density ratio of the liquid and vapor phases.
For the high
density ratio of sodium at atmospheric pressure, the code
fails to converge for conditions of boiling inception and
condensation.
For the present the THERMIT calculation results are not
in good agreement with the experiment data, but the general
tendency and the order of magnitude of the results are right.
THERMIT
is
a powerful tool in
the sense that it
easily accept the developments that will be made in
can
the
-135-
future.
If more studies on the numerical and physical
models would be done, THERMIT might be improved to the state
that it can handle a wide range of real LMFBR accident con-ditions.
_
___~~__
Ilj
-136-
Chapter 7.
1.
Recommendations for future work
Improvement of the physical models in THERMIT
Problems discussed in Chapter 3 will have to be solved.
a. Forced and natural convection correlations
b-. Flow regime dependence
c. Condensation modeling
Interfacial mass transfer
Interfacial energy transfer
d. Geometry dependence
2.
Validation of the physical models against experiments
More experiments should be done to test the physical
models.
3.
Improvement of the overall numerical scheme in THERMIT
The code failed at a high density ratio of liquid and
vapor phases in the present application.
4.
Improvement of the loop simulation method
Implicit numerical scheme may be used to hold the plenum
conditions more accurately.
It is also possible to keep the plenum conditions by a
large heat transfer through the structural material.
-137-
5.
Mathematical model for the oscillation in a two-phase
loop experiment
More general and complete explanation of the two-phase
oscillations should be done.
THERMIT may be a helpful
for this work.
6.
Modification of THERMIT momentum conservation equations
to accept a variable flow area in axial direction.
.-~---- -- -
111
-138-
References
[1]
Reed, W.H.,
Stewart, H.B.,
"THERMIT, A Computer pro-
-gram for Three-dimensional Thermalhydraulic Analysis
of LWR Cores," EPRI NP-2032 Project518, Final Report,
Sep. 1981.
.[2]
Garrison, P.W. et al, "Dryout Measurements for Sodium
Natural Convection in a Vertical Channel," ORNL/TM-7018
[3]
(December 1979)
Costa, J. and Charlety, P.,
"Forced Convection Boiling of
Sodium in a Narrow Channel," in Liquid Metal Heat Transfer
and Fluid Dynamics, ASME winter annual meeting, New York,
Nov. 1970.
[4)
Wilson, G.J.,
"Development of Models for the Sodium
Version of the Two-phase Three-dimensional Thermal-hydraulics Code THERMIT," M.S. Thesis, Department of
Nuclear Engineering, MIT, (1980)
[5]
Autruffe, M.A.,
"Theoretical Study of Thermohydraulic
Phenomena for LMFBR Accident Analysis," M.S. Thesis,
Department of Nuclear Engineering, MIT, (1978)
[6]
Collier, J.G.,
Convective Boiling and Condensation,
McGraw-Hill International Book Company,
2 nd
ed.,
(1980)
[7)
El-Wakil, 11.M.,
Nuclear Heat Transport, International
Textbook Company, Scranton, Pa.,
(1971)
-139-
[8]
Tang, Y.S.,
Coffield, R.D., Markley, R.A.,
Thermal Ana-
-lysis of Liquid Metal Fast Breeder Reactors, American
Nuclear Society,
[9]
Seiler, J.Y.,
(1978)
"Sodium Boiling Under Single Channel NaCalculational Results
-tural Convection Conditions;
and Analysis of ORNL/SBT Experimental Results," AIChE,
NewOrleans, Nov 8-12, 1981
[10]
Lahey, Jr R.T.,
"Nuclear System Safety Modelling," in
Nuclear Reactor Safety Heat Transfer, O.C. Jones, editor
Hemisphere McGrawHill, 1981
[11] Quandt, E.R.,
-tions,"
"Analysis and Veasurement of Flow Oscilla-
Chem. Eng. Progr., Monograph and Symposium Seri-
-es No. 32S, p. 1 1 1 .
[12] Wissler, E.H.,
Isbin, H.S.,
Amuneson, N.R.,
"Oscillatory
Behavior of a Two-phase Natural-circulation Loop," Am.
Inst. Chem. Eng. J.,
[13] Wallis, G.B.,
Flow Systems,"
3(1956)2.
Heasley, J.H.,
"Oscillations in Two-phase
J. Heat Transfer, Trans. ASME Series C,
83(363)369.
[14] Levy, S.,
Beckjord, E.S.,
"Hydraulic Instability in a
Natural Circulation Loop with Net Steam Generation at
1000 psia," ASME Paper 60-ET-27, 1960.
[15]
Personal communications with Dr. Allan Levin in Oak
Ridge National Laboratory.
--
Hi
ll
illilm
imol
,,
-140-
Appendix A.
Calibration of heat input
Outer wall temperature data of experiment in adiabatic
zone
107R2 .
108R2
100P.3
100R2
100R4
TE 124
7630C
864 0 C
9070C
873 0 C
8450C
TE 125
754
856
898
863
S48
TE 126
756
865
906
871
870
TE 127
754
873
900
866
875
TE 128
763
884
893
856
877
TE 129
763
881
881
845
874
TE 130
762
875
886
852
880
759.3
871.1
895.9
860.9
867.0
AVERAGE
-
The temperature difference between sodium and outer wall is
about 11C.
Therefore the sodium temperature is calculated
as follows.
107R2
300W
759. 3 + 11 = 770.30C
108R2
500W
871.1
+ 11 = 882.1
100R3
700W
895.9
+ 11 = 906.9
100R2
700W
860.9 + 11 = 871.9
100R4
800W
867.0
+ 11 = 878.0
-141-
Test No.
Power given in
AT
AT(calculated
ORNL/TM-7018
(measured)
with 70% power)
107R2
300 W
350.30C
279.65 0 C
108R2
500
462.1
362.35
100R3
700
486.9
455.55
100R4
800
458
472.95
Since the total heat input appears as the enthalpy increase
of the single-phase liquid sodium,
p
actual power
input
m =
measured
volumetric flow rate
measured
measured
(density)x(
where
c
P
= 1447 J/kg OC
As a result of the calculations,
Test No.
Power given in
70% Of the
Actual
ORNL/TM-7018
power given in
power
ORNL/TM-7018
input
107R2
300 W
210 W
263.3 W
108R2
500
350
446.6
100R3
700
490
522.9
100R4
800
560
541.1
m
ll
iUnIai
i InH
i-
I
i
lllngllll
i i
-142-
0.7
0.6
05
0*
04 -
0
1.
0
02
0.4
0.6
0.8
10
1.2
1.4
1.6
18
2.0
iTS (kW)
?i
Al.
Relative error in test section power determination
as a function of test section power
l.inil
illi lili .
iilliHI
I
-143-
actual power
input
[W]
Error band according
800
to ORNL/TM-7018
600
----
power calculated
With temperature
400
data in ORNL/TM-7018
200
power reported
0. 0
0.0
Fig A2.
200
400
600
800
Comparison of the actual power and the reported
power in ORNL/TM-7018
[W]
lull
I
UlllIIlill
.
I
-144-
Energy conservation of THERMIT
Appendix B.
THEPMIIT conservation equations for 1-D steady-state
Mass
- (ap v
z) =
r
9 zi[ (l1-a)pkv~z
Momentum
(p
-z
-
Energy
2
vvz
(B1)
=
+ aaP
(B2)
-r
=
-Fwv
wV
-F. 1
8
2
[ (1-a)p v 2 ]] + (1-a) +P - -Fwk
k z
-z
-- (ap e v
) + P aZ-(av
vz
+ P-[(1
az@[ (l-a)p.ke kv z ]
aZ
(B3)
-cpvg
vZ
+F i
-
(1-a) P g
(B5)
) = QwV +Qii
- a
z
)v Ez
(B4)
= Qwt -Qi1
(E6)
First law of thermodynamics
S
z.
zvzt
vvz(h
v + 2
+gz Z)
= Qwv
1
WV +Qi
(B7)
2
V".
-
(1-a) pv z(h
zz
= Qw
-Qi
(B8)
Reexpressing Eq B5 and B6 in terms of enthalpy,
(I(ao h v
z
v v vz
)
Pz
vz z
Qwv
+Qi1
wV
(B9)
lmlil lill,
k
-145-
(-)
ah
v
a-Z[(l-a)h
z] -
P
z
(cvz)a
=
-Qi
Qw
(B10)
Multiplying Eq B3 and B4 by the respective phasic velocities,
the mechanical energy conservations are obtained as follows.
aP
= v
-(-a)v
p
-avZ
3
(ap vV
2
)
+ (Fw +F i +aOPg )vz
+ IF
-a)p2
a
-(l-)V~z - = Vtz[
(l-)
£Vz
(Bll1)
-F
w
i
+(1-a)p gz IV
(B12)
The THERMIT energy conservation equations are obtained by
substituting Eq B11 and B12 into Eq B9 and B10.
Vvz
3
2
(aPvV)
and
The terms
and their liquid counterparts are
pvg
transformed through the following procedure.
vzaz
a 1(
- 2
v vz
v vz
3
v vz
ap Vv Vz
2
z
vz)(pvVvz
vz
vz az
av vz
2
=vv
v vz
z +
5z
v
2
v vz
2
vz
2
a
z
v
v(Vz
a ( vvz )
vVvz
Z(
(314)
Therefore
2
vz
2
a
- (.a vvz ) =
1
z 2
V3
v v
z) +
a
vz
2 az (apv v vz )
(15)
-146-
Utilizing Eq El,
Vvz F
Eq B15 reduces to
r 2
Vvz
3
~a
2
v vz
2 Pvvz
+
(B16)
Similarly for the liquid, .utilizing Eq E2,
=
z
V
)p
Vz [-(1-
3
1
2
"
-[Y(l-a)p pVz
2
V2z 8
2 8z [ (l-)p
S[1
=z 2
-
3
)zV ]
Viz]
F 2
-
f
(317)
z
Similar procedure is applied to the potential terms in Eq
B7 and B8 so that the following equations result.
v
Spg
z)
z ov9z vz
ap g v
(apvg
=
z)
v
z
z
- z-(ap g v )
v z vz
az
- zg
r
z[ S-
(B 18)
gz
9z (1l-a)p kg zvVz z] + zg zF
-
Utilizing the results of Eq B 16, BE17, B 18 and B 19 in
in combination with Eq B 9, B 11 and B 10, B 12 yields,
( 19)
-147-
-- a
S(h
vz
2
v(h v +- 22 +g Z z)] +f-v
2 vz
vvzvz
8z
= Q
(h
'z
+T
[(1-a)p v) (h
-zg
z
P +(F wV +F.)v
1
VZ
+Qi
k120)
+g z)]
-
r 2
V
+zg r +(F
-F )v~z
= Qwk -Qi
(B21)
Finally upon rearrangement of Eq D20 and B21, the THERMIT
energy conservation equations are obtained.
V
vz
+(h
oZ zap v v vz v
2
(h
+g
v2
+Q
Q
wv
i -Vvz +zg
zz)=
-(FW
[ (l-a)
p v(h
+
2
+g
z)]
-(Fwk
+F
=
z
(B2:2)
)v
Q
-Q
+- v
-zgz
-Fi)Vz
When Eq B22 and B23 are compared with Eq B7
(B23)
and BS, it
can be
seen that the present THERMIT energy conservation equations
are in contradiction with the first law of thermodynamics.
-148-
Appendix C.
Plenum temperature
dhZ2, dhv2
!
#*
1
dh£1,
Fig Cl.
dhvl
Plenum energy conservation
Q
-149-
VpTCp (T-To)
Q = dhZ1 + dhZ2 + dhvl +dhv2 + V)
(Ci)
where
Q : the total amount of heat that should be extracted from
the plenum at time step n to keep the temperature at To.
V : volume of the plenum
CT : specific heat of sodium at temperature T
T : sodium density at temperature T
T : the plenum temperature at time step n
To
: the desired plenum temperature at time step (n+!)
(tn+l-tn)
At : time step size
dhl1 : the amount of heat that should be extracted from the
plenum to keep the in-flux sodium at bottom face at
temperature T.
dhZ2, dhvl, dhv2 : similarly defined as dhl1.
dhZl(or dhZ2,
dhvl, dhv2)
is
equal to
zero if the flow is in outward direction.
The scheme of Eq Cl worked very well for single-phase
cases, but in two-phase case where there is condensation in
the plenum, the error in the plenum temperature was not
tolerable.
Hence for the two-phase simulated 1-D loop
analysis, an iterative method is used to get a constant plenum
temperature.
-150-
heat extracted
from the plenum
Q1
Q4
Q3
-
T1
Fig C2.
To
4.
T2
Iterative'scheme
plenum temperature
-151-
In Fig C2, a parabola is drawn with (T1, Q1)
(T 3 ,
(T 2 , Q2 ),
Q3 ) and Q4 is calculated so that it corresponds to To
on that parabola.
With Q 4, the whole calculation is repeated
and a new temperature T 4 is obtained.
Another parabola can be drawn with (T 2 ,
and (T4, Q4 )
Q2 ),
(T 3 ,
Q3 )
and the same procedure is repeated until a
satisfactory plenum temperature is obtained.
-152-
Appendix D.
Dependence of the wall friction on the void
fraction
Autruffe's wall friction correlation
For liquid
F0.18
Z= 2De [Re ]
0.18
F
= 2De
Re]
-0.2 Pu
P uI
-0.2
pu
lu
a < Odry
d ry
u
1-aldry
>dry
For vapor
0.18
v
2De
-0.2
[Rev
(Dl)
-dry
Pv Uv Uv-a
(D2)
(D3)
dry
For a < adry' only the liquid is in contact with the wall.
FT = Fk
f = 0.18[Re]0.2
(D4)
The mass flux G is expressed as follows,
G =
(l-a)p £u
= P£u£[(1-a)
Since pv/pz
1/2000,
+ aP uv
+
P£Sa]
S = u /u
(D5)
-153-
G = pu z(1-a)
Re
=
Therefore Re
constant.
(l-alp u De
(D6)
G De
(D7)
is independent of the void fraction if G is
It means that the friction factor f is also indepen-
-dent of the void fraction a for constant G.
For a 2dry'
FT = Fk + F
Table Dl shows that the liquid wall friction is dominant
over the vapor wall friction for a < 0.9999.
Hence it may be
said that f is independent of the void fraction for constant G
unless the flow is almost single-phase vapor.
-154-
Table D1
Wall friction forces by the liquid and vapor phases
in contact with the wall
F/G
1.8
Fv/G
8
1.G
FT/G
1.8
0.1
0.0254
0.0
0.0254
0.2
0.0321
0.0
0.0321
0.3
0.0419
0.0
0.0419
0.4
C.0570
0.0
0.0570
0.5
0.0821
0.0
0.0821
0.6
0.1282
0.0
0.1282
0.7
0.2277
0.0
0.2277
0.8
0.5111
0.0
0.5111
0.9
2.0312
0.0
2.0312
0.95
8.0199
0.0
8.0199
0.957
10.798
0.0
10.798
0.96
11.58
0.03
11.61
0.97
15.28
0.05
15.33
0.98
0.99
22.44
42.17
0.10
0.32
22.54
42.49
0.995
74.94
1.00
75.94
0.997
107.88
2.16
110.04
0.999
178.77
8.65
187.42
0.9995
189.70
16.01
205.71
0.9999
106.50
32.47
138.97
0.99999
20.76
40.03
60.79
0.0
41.03
41.03
1.0
1.0
where
De
= 3.25x10
-3
= 1.6476x10
S
m
-4
kg/m s
kg/m s
SJ = 1.9644xi0-5
1.9644x10
kg/mn s
p
2
= 742.18 kg/m 3
= 0.27 kg/m
3
-155-
Appendix E.
Typical experimental data
Some of the typical data from ORNL/TM-7018 are presented
here.
These are all stable boiling data and Fig E4 and E6
show the void propagation clearly.
This void propagation
mode can be compared with the simulated 1-D loop results in
Fig 4.10 -
4.19 in
Chapter 4.
-
I III-
YYIYII
-156-
w
DH_
<G
U.J cn
LEGEND
LU
w
-0
H
IE 117
TE 118
TE 119
0
Cl
LEGEND
D- PE-2B
0- FT-7
LU
.J
O
z
0
-J
-I
t-
0
LEGEND
0 - FE-5B
o
u'3--
0
LL
U
z
LEGEND
0- FE-16
60
I
I
70
80
I
90
1
100
I
110
I
12
TIME (s)
Fig El.
Stable boiling of the test run 129R1 at 1.3 kW
(taken from ORNL/TM-7018)
-157-
w
DU
U~j
LEGEND
o- TE 117
o- TE 118
6- TE 119
w
H
C_
2-
8
LEGEND
0 - PE-2
jJJJ1
JJJJJ
J
o- FT-7
LEGEND
o- F'-5B
- -
o
P1v
VVAVVA
LE- fND
o - FC-IB
TIME (s)
Fig E2.
Stable boiling of the test run 127R1 at 1.5kW
(taken from ORNL/TM-7018)
~---~ -
~
- ~rrrslrr~l~
-158-
0
"rl"~~t
mit
'lIpII~r71R
LEGEND
0- CE 4-40
I
1
1
-
LEGEND
0- CE 4-39
1nTr
V,7
LEGEND
0- CE 4-38
Lfl1
-J
0
Ln,
._4
.. J
LEGEND
0- CE 4-37
0
10
20
30
TIME (s)
Fig E3.
Void detection system data for the test run 127R1
(taken from ORNL/TM-7018)
m3
O
CD
..j
I
0
0
t
0l
(D
t-t
CD
O-,
0
O
F-j
rF
r(D
F
0O
(D
c1
(D
H.
Lr
:j
rt
O
0
C(D
r
Fl
0(D
O
Y
I-
(D
(n
,"o
-5.0
~)
0.0
INLET FLOW
(rnI/s)
5.0
iJ
-11
-C
OUTLET FLOW
(min
l/s)
5.0
Cir
rTj
WG
r\.
-v
INLET PRESSURE
(kPa)
5
0
Z
O
C-
VOID
24
DETECTOR
-160-
0
0
r"
D
0-
LU
uJ
OW %IA
.
LEQNO
TE 117
TE 118
TE 119
LEGEND
S- PC.-2B
o- PT-7
E,.
C
o-
LE END
FE-SB
o
U
i
LEGEND
o- FE-l8
c!
,
TIME (s)
Fig E5.
test
Stable boiling at high
flow and
low quality
run 120R1 at 1.5 kW (taken from ORNL/TM-7018)
for
the
-161-
0
wC
0
I BOILING
uj
V)
LOl
-
-~
_J
LEGEND
o- PE-2
z
I
I
I
i
i
LEGEND
o - FE-SB
I
L;?
z
LEGEND
0- FE-2B
L4 j
TIME (s)
Fig E6.
Relationship between voiding pattern, flow and
pressure for the test run 120R1
(from ORNL/TM-7018)
-162-
Appendix F.
Typical computer inputs and outputs
Some of the important computer inputs and outputs are
included here.
The inputs are for the test run 129R1 and the
simulated and actual 1-D loop analysis.
The outputs are from
the ORNL/TM-7018 test section calculation of the single phase
run 107R2 at 210 W with 4-Eq model, the two-phase run 129R1
at 910 W with HEM, 4-Eq and 6-Eq models.
The French forced
convection results are also listed for the mass flow rate and
tube diameter 0.014 kg/s, r = 0.002 m and 0.013 kg/s, r =
0.003 m.
The single-phase results of the simulated and actu-
-al I-D loop calculations are also shown.
Note that the actual
1-P loop calculation has a two-dimensional geometry from the
computational point of view, therefore it is composed of four
channels.. Finally the simulated 1-D loop calculation results
with the vapor density 100 and 200 times increased are listed.
Note that these are the results at the inception of boiling
and a reverse flow is indicated at the inlet of the test sec-tion with the vapor density 100 times increased.
Input for the test section steady-state run 129R1 with 4-Eq model
2
ORNL- dryout measurement for sodium natural convection
SBTF
in a vertical channel
$intgin nc=1 nz=10 nr=1 narf=1 nx=O nrzs O ihtf=1 thts=O
iss=1 ixfl=O idump=l ibb=1 iwft=O ichnge=O ihtrpr=1
ishpr=1111 nitmax=3 ipfsol=10 neq=4 ieqvax=l ieqvtr=1 ntabls=l ipowt=1 Ihtrpr=l
$realin epsn=l.e-3 hdt=3.25e-3 pdr=1.2533 hdr=1.Oe+10 rnuss=7.0 radf=i.625e-3
$
delpr=0.00001 delro=0.00001 delem=0.00001
$
$rodinp q0=910.
$ ncr
1
indent
$
O
ifcar
$
1
$ nrzf
1
$ nrmzf
1
mnrzf
$
4
$ dx
4.07327e-3
$ dy
4.07327e-3
dz
$
6(0.194) 6(0.3)
$ arx
10(0.0)
ary
$
10(0.0)
$ arz
11(8.295768e-6)
$ vol
5(1.609378992e-6) 5(2.4887304e-6)
$ hedz
3.25e-3
$ wedz
3..25e-3
$ p
6(1.226e+5) 6(1.01e+5)
$ alp
12(0.0)
$ tv
12(693.15)
vvz
$
11(12.054e-2)
twF
$
10(693.15)
qz
$
5(1.0) 5(0.0)
qt
$
1.0
$ qr
1.0
rn
$
1
$ drzf
1.625e-3
3
0.0 0.0 50.0 1.0 1000.0 1.0 $tablel
Stimdat tend=-1.0 dtmin=1.0e-8 dtmax=5.0
$
iredmx=25
dtsp=10.0 dtlp=30.0
0
-163-
$
Input for the simulated i-D loop analysis with 4-Eq model
Input for 1-0 loop analysis
$intgin nc=i nz=26 nr=l narf=1 nx=O nrzs=O thtf=1 ihts=O iss=2
ixfl=O Idump=1 iflash=l itb=O ibb=l twft=O ichnge=O ishpr=l11
nitmax=3 ipfsol=iO neq=4 leqvax=O ieqvtr=1
$
$realin epsn=1.e-4 hdt=3.25e-3 pdr=1.2533 hdr=i.Oe+iO rnuss=7.0
radf=1.625e-3 delpr=l.e-7 delro=i.e-7 delem=i.e-7
$
$rodinp q0=210.
$
1
$ ncr
indent
$
0
$
ifcar
1
$ nrzf
1
1
$
nrmzf
4
$ mnrzf
4.07327e-3
$
dx
4.07327e-3
$
dy
$
dz
2(0.6175) 5(0.193) 5(0.3) 16(0.617 5)
arx
$
26(0.0)
$
ary
26(0.0)
arz
$
27(8.295768e-6)
5.1226368e-6 5(1.6010832e-6) 5(2.41887304e-6) 15(5.1226368e-6) $ vol
$
hedz
3.25e-3
wedz
$
3.25e-3
28(1.29039e+5)
$ p
28(0.0)
$
alp
28(863.15)
$
tv
$
vvz
27(8.0e-2)
26(863.15)
$
twf
$
qz
-210. 5(1.0) 5(0.0) 0.0 14(0.0)
1.0
$ qt
1.0
$ qr
1
$ rn
1.625e-3
$ drzf
$
$timdat tend=-1.0 dtmln=l.e-8 dtmax=5.0 dtsp=5.0 dtlp=lO.O iredmx=25
0
-164-
Input for the actual 1-D loop analysis with 4-Eq model
1
Input for loop analysis of 129ri - 4 channels
$intgin nc=4 nz=14 nr=l narf=l1 nx=O nrzs=O Ihtf=i ihts=O iss=2
ixfl=1 idump=l iflash=l itb=O ibb=1 iwft=l ichnge=O
ivelpr=1 ishpr=1111 nitmax=3 ipfsol=22 neq=4 $
$realin epsn=l.e-4 hdt=3.25e-3 pdr=1.2533 hdr=1.0e+10 rnuss=7.0
radf=1.625e-3 delpr=1.e-6 delro=l.e-6 delem=i.e-6
$
$rodinp q0=910.
$
4
$ ncr
$
indent
0
1
$
ifcar
,
$ nrzf
1
i
$ nrmzf
$ mnrzf
4
$ dx
8.01106e-2 2(0.915) 4.07327e-3
$ dy
4(4.07327e-3)
$
dz
3(8.01106e-2) 5(0.194) 5(0.3) 3(8.0110Ge-2)
$
arx
14(0.0) 3(2(4.07327e-3) 10(0.0) 2(4.07327e-3))
$ ary
56(0.0)
1.0e-12 14(3.26312e-4) 2(i.0e-12 3.72704e-3 11(1.Oe-12) 2(3.72704e-3))
$
arz
1.0e-12 14(1.659153e-5)
2(2.614106e-5) 5(6.330455e-5) 5(9.789363e-5) 2(2.614106e-5)
2(2(2.985756e-4) 10(1.0e-12) 2(2.985756e-4)) 2(1.329157e-6)
$ vol
5(3.218757e-6) 5(4.977459e-6) 2(1.329157e-6)
4(3.25e-3)
$ hedz
$
wedz
4(3.25e-3)
4(8(1.2e+5) 8(1.Oe+5))
$ p
$
alp
64(0.0)
$
tv
64(860.)
$ vvz
GU(O.uy
$
twf
56(860.)
qz
2(-210.) 5(1.0) 5(0.0) 2(-700.) $
$ qt
3(0.0) 1.0
$ qr
1.0
$ rn
20 2(225) 1
$
drzf
1.625e-3
$
$timdat tend=-1.0 dtmin=1.Oe-8 dtmax=1.0 dtsp=0.5 dtlp=5.0 iredmx=25
0
-165-
Output of the test section steady-state run 107R2 with 4-Eq model
time Step no =
36
number of newton Iterations
number of
inner
time =
1
iterations .
total reactor power =
total heat transfer =
flow enthalpy rise =
34.780457 sec
time step size w 0.96364E+00 sec
=
0
0
Inlet flow rate outlet flow rate u
0.210
0.210
0.210
-
maximum relative changes over
in pressure:
in mixture density:
in mixture energy:
Ic
Iz
P(bar)
.2113
.2032
.1952
.1873
.1794
.1715
.1638
.1561
.1484
.14Q9
.1333
.1257
.11130
.1103
.1026
.0949
.0872
.0794
.0717
.0640
.0563
.0486
.0409
.0331
.0254
.0177
.0100
void
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
% qual
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
hm
913556.
948823.
984091.
1019358.
1054625.
1009892.
1125160.
1160427.
1195694.
1230961.
1266229.
1266228.
1266227.
1266226.
1266225.
1266223.
1266222.
1266221.
1266220.
1266219.
1266218.
1266217.
1266216.
1266213.
1266206.
1266181.
1265920.
rom
T vap
T liq
850.63
844.24
037.83
831.40
824.93
818.43
811.91
805.37
798.81
792.24
785.66
785.66
785.66
785.66
785.66
785.66
785.66
785.66
785.66
785.66
785.66
705.66
785.66
785.66
785.66
785.66
785.66
693.15
720.98
748.82
776.72
804.68
832.68
860.71
888.76
916.80
944.83
972.83
972.83
972.83
972.83
972.83
972.83
972.83
972.83
972.83
972.83
972.83
972.83
972.83
972.83
972.82
972.80
972.80
693.15
720.98
748.82
776.72
804.68
832.68
860.71
888.76
916.80
944.83
972.83
972.81
972.03
972.83
972.83
972.83
972.83
972.83
972.83
972.83
972.83
972.83
972.83
972.83
972.82
972.80
972.80
-166-
0.001 kg/s
0.001 kg/s
the time step
0.0000
0.0000
0.0000
T sat
1178.99
1178.22
1177.45
1176.68
1175.92
1175.16
1174.40
1173.64
1172.88
1172.13
1171.38
1170.61
1169.83
1169.05
1168.26
1167.46
1166.67
1165.86
1165.06
1164.24
1163.43
1162.61
1161.78
1160.95
1160.11
1159.27
1158.42
vvz
vlz
0.084
0.085
0.086
0.086
0.087
0.088
0.088
0.089
0.090
0.091
0.091
0.091
0.091
0.091
0.091
0.091
0.091
0.091
0.091
0.091
0.091
0.091
0.091
0.091
0.091
0.091
0.084
0.085
0.086
0.086
0.087
0.088
0.088
0.089
0.090
0.091
0.091
0.091
0.091
0.091
0.091
0.091
0.091
0.091
0.091
0.091
0.091
0.091
0.091
0.091
0.091
0.091
roV
0.49
0.32
0.32
0.32
0.32
0.31
0.31
0.31
0.31
0.31
0.30
0.30
0.30
0.30
0.30
0.29
0.29
0.29
0.29
0.29
0.28
0.28
0.28
0.28
0.28
0.27
0.31
rol
850.63
844.24
837.83
831.40
824.93
818.43
811.91
805.37
798.81
792.24
785.66
785.66
785.66
785.66
785.66
705.66
785.66
785.66
785.66
785.66
785.66
785.66
785.66
785.66
785.66
785.66
785.68
Cutput of the test section steady-state run 129R1 with HEM model
I i -i,
.. t -'I
nu.,ther
nu iter
totI
I
,ui)
I
:
fl tI,
time
./.
1
1
n. atn it ter t ions =
rr,:r" ite rati ors =
o.
) I
re
Z(
j*r
t r
r
It-y
riF
=
=
in (et
outlt et
naixiinur
ic
i ;
I'( bj r)
voi
U
L)U
L..Ii(I UOII
'.'
/7 ,"21
935
17 '
.
II..15'1. :
:.741
1', i,.97 ),)
( .96113
. 5'
t'.9 ) -')
.;t
*~
1
1,
! . j I t!
1 S
ti.;o
(1. 91 1 1,
.
lua I
I. OUO
0 (100
,. 7
7
5.0'I1
5.427
6.442
/.29
time
stel
size
= d.390165D-02
sec
d
I1
0. 910
0i.91t
U.,98
.rrc |
I S ('
sec.
14.'7109l
ron.
iit
913550.
11275 1'.
1341 46 .
1555 2 70.
1 769119 3.
1 ' ;l 2 ,.
1 ,/$1 ,9 5.
19 7? 750).
197/143.
1973425.
1907437.
18'0)1 161.
If (6w
rate
ft ow ra t e
rel at ive
cihaiges
pressure:
in
in
mixture densi t y:
mixture energy :
T vap)
T Ii(
693. 15
693.
35U.63
862.
811.54
4,62 .32
77 1.66 11)32.12 1i)32.
731 .71, 1 199. 2 1199.
1331.
1 05.2? S13 1. 7,
19.03 1331.16 1331 .
1321.141 1321.
17.0'
14.71 1 307.70 13017.
12.21 1 2911.4 . 1290.
9.56 12?67.155 1261.
0.01 1 233.71W 1233.
6.44 1233.70 1233.
-167-
over
in
0.011
ku/s
0.11
k;/s
the time
0.0(00UU6
0.000U010)
0.0000(IJU
T sat
1334.10.
1333.4 1
1332.89
1332.33
1331.79v
1331.16
13 2 1 .41
1307. 7U
1290.48
1267.5 5
1233. 7U
115 8.42
steo
vVZ
0.121
0.126
0.133
0. 141
5.3 )2
6.0U6
6.917
b.4J2
10.738
15.394
vtz
U.
0.
0.
0.
U.
5.
L.
6.
t.
1(.
15.
121
126
133
1 IU
Y75
392
977
402
733
394
Output of the test section steady-state run 129R1 with 4-Eq model
time step 1no =
number
humber
total
total
flow
1.99
of ne(' ton i teralt ions
of inner i terat ions
=
time =
13.014007
1
1
0
0
0.910 kW,
0.910 kW
0.900 kW
reactor ,other
heat trdnsfer
enthatoy rise =
sec
time
intet
outlet
maximum
step
flow rate
flow rate
size
=
over
in mixture densi ty:
Sin
ic
iz
P(,ir)
void
% aual
1. 7o0t4
1. 744 39
0.0i00)U0. 000
0.000
tO
0.000
1.71 5i' ,
1 .90'?53
1 .6723 B
11.
. 5901"9
512
7
1. 3 9':
L. 00 0l)
0 . 8 )( 3
0.0000
1. 26433
1 . 14591
1.01
0
0. )413
i. 9 4 0
(0.94147
(I.?467
(0. 94 ,4
U.9515
V.9515
0.00%0
0. 000
3.
6.
B.
t.
.
753
066
214
406
620
9. 863
9.141
ro1i
91 356.
1127510.
1341374.
1554799.
1606647.
1625322.
1616518.
1604648.
1591510.
1576163.
155 6 15.
1554918.
850.
811.
771.
7 1.
75.
43.
41.
40.
39.
38.
36.
36.06
-168-
mixture
T
vap
693.
862.
032.
199.
219.
217.
211.
2103.
104.
18 4.
172.
1172.03
energ y:
1
i (I
693.
862.
1032.
1199.
1219.
1217.
1211.
1203.
1194.
1184.
1172.
1172.63
0.84047D-02
0.001
0.001
=
relative
changes
in pressure:
=
T
sec
ky/s
kg/s
the time step
0. 000002
0.00UU001 0
0. 000002
sat
1223.77
1222.60
1221 .48
1220.41
1219.38
1217.39
1211.28
1203. 34
1194.4 5
1184.31
1172.63
1158.42
v VZ
0.121
0.126
0.133
0.140
9. 72'3
20.210
21.511
23.345
25.5)2
28.450
32.12-5
vtz
U.121
0.126
0.133
0.140
1.316
2.207
2 .268
2.326
2.4 00
2.464
2.604
Output of the test section steady-state run 129R1 with 6-Eq model
time =
18.743907 sec
time step no - 1749
number of newton iterations
1
0
0
number of inner iterations =
total
total
reactor power =
heat transfer =
flow enthalpy rise
=
0.910
0.910
0.900
time step size
inlet flow rate =
outlet flow rate -
=
0.83948D-02 sec
0.001 kg/s
0.001 kg/s
maximum relative changes over the time step
0.000000
in pressure:
0.000000
In mixture density:
0.000000
in mixture energy:
Ic
iz
P(bar)
void
% qual
1.76140
1.74495
1.72926
1.71434
1. 70010
1.67289
1 .59138
1.49026
1.38313
1.26852
1. 14603
1.01000
0.0000
0.0000
0.0000
0.0000
0.8970
0.9413
0.9431
0.000
0.000
0.000
0.000
3.763
8.076
8.230
8.422
8.637
8.880
9. 159
0.9448
0.9468
0.9484
0.9516
0.9516
rom
913556.
1127520.
1341483.
1555441.
1606887.
1625660.
1616634.
1604755.
1591609.
1576251.
1559693.
1554988.
T vap
T liq
693.15
693.15
862.57
862.57
1032.78 1032.78
1200.25 1200.25
1219.50 1219.52
1217.61 1217.62,
41.87 1211. 32 1211.32
40.72 1203.38 1203.38
39.37 1194.49 1194.49
38.23 1184.34 1184.34
36.05 1172.66 1172.66
36.02 1172.66 1172.66
850.63
811.48
771.51
731.55
75.27
43. 11
-169-
T sat
1223.81
1222.64
1221.52
1220.45
1219.42
1217.43
1211.32
1203. 37
1194.48
1184.33
1172.64
1158.42
vvz
0.121
0.126
0.133
0.140
9.743
20.222
21.535
23.372
25.621
28.483
32.163
vlz
0.121
0.126
0.133
0.140
1.318
2.208
2.268
2.327
2.400
2.464
2.605
rov
rol
0.7139
0.5929
0.5111
0.4508
0.4416
0.4350
0.4155
0.3911
0.3651
0.3371
0.3069
0.2704
850.63
811.48
771.51
731.55
726.92
727.37
728.89
730.80
732.94
735.38
738.18
738.18
Output of the French experiment with m = 0.014 kg/s and r = 0.002 m
time step no = 1231
time =
2.486363 sec
number of newton Iterations =
I
numbe r of inner iterations =
1
0
0
total reactor power =
total heat transfer =
flow enthalpy rise =
time step size = 0.13904D-02 sec
inlet flow rate =
outlet flow rate =
10.000 kW
10.000 kW
9.969 kW
0.014 kg/s
0.014 kg/s
maximum relative changes over the time step
in pressure:
0.000000
in mixture density:
0.000001
in mixture energy:
0.000000
Ic
iz
P(bar)
2.92212
2.91222
2.89237
2.87289
2.85305
2.77086
2.67272
7.54072
2.35472
2.07823
1.64716
1.20000
void
% qual
hm
0.0000
0.0000
0.0000
0.0000
0.5822
0.6629
0.7166
0.7639
0.8079
0.8463
0.8961
0.8961
0.000
0.000
0.000
0.000
0.244
0.347
0.473
0.648
0.908
1.327
2.081
955524.
1137047.
1318567.
1500088.
1676179.
1673517.
1669365.
1662942.
1652588.
1633902.
1600630.
1593772.
rom
T vap
843.05 726. 15
869.99
809.75
775.92 1014. 13
742.01 1156.67
297.33 1286.95
240.35 1282.93
202.48 1278.01
169.20 1271. 17
138.21 1261.03
111.26 1244.74
75.98 1215.53
75.88 1215.53
-170-
T liq
T sat
726.15
869.99
1014. 13
1156.67
1286.95
1282.93
1278.01
1271.17
1261.03
1244.74
1215.53
1215.53
1290.26
1289.79
1288.84
1287.91
1286.95
1282.93
1278.01
1271.17
1261.03
1244.74
1215.53
1177.91
vvz
viz
1.300
1.353
1.300
t.353
1.412
1.412
S1.477
1.477
6.468
8.288
10.800
14.549
20.667
3.682
4.552
5.400
6.454
7.885
9.758
14. 194
32.342
59.334
rov
1.1393
0.7243
0.7197
0.7153
0.7108
0.6919
0.6694
0.6390
0.5959
0.5312
0.4289
0.3125
rol
843.05
809.75
775.92
742.01
710.63
711.60
712.79
714.45
716.90
720.84
727.88
727.88
Output of the French experiment with m =
time step no = 1332
time =
3.129023 sec
ions number of newton I ternt
niumber of inner iterations =
1
0
0
total
reactor power
=
total heat transfer =
flow enthalpy rise =
0.013 kg/s
time step size
inlet flow rate =
outlet flow rate =
10.000
10.000
9.930
and r
=
=
0.00'3 m
0.151170-02 sec
0.013 kg/s
0.013 kg/s
maximum relative channges over the time step
in pressure:
0.000000
in mixture density:
0.000001
in mixture energy:
0.000000
Ic
iz
P(bar)
2.20)11
2.19544
2.18253
2. 17014
2. 15810
2.04965
1.93596
1. 81016
1 66862
1.50620
1.31862
1.20000
void
% qual
0.0000
0.0000
0.0000
0.0000
0.8857
0.8977
0.9036
0.9094
0.9156
0.9215
0.9308
0.9308
0.000
0.000
0.000
0.000
2.405
2.574
2.759
2.974
3. 30
3.546
3.946
955524.
1152175.
1348818.
1545428.
1649570.
1643226.
1633963.
1622899.
1609605.
1592626.
1572229.
1569407.
rom
T vap
843.05
806.93
770.24
733.56
83. 12
74.24
70. 12
66. 13
61.78
57.71
51.12
51.09
726.15
882.09
1038. 11
1191.90
1249.61
1242.96
1235. 68
1227.21
1217. 12
1204.65
1188. 85
1188.85
-171-
T liq
T sat,
796.15 1252.24.
882.09 1251.84
1038. 11
1191.90
1951.07
1950.33
1249.61 1249.61
1242.96 1242.96
S235.68 1235.68
1227.21 1227.21
1217. 12 1217. 12
1204.65 1204.65
1188.85 1188. 85
1188.85 1177.91
vvz
vlz
0.533
0.557
0.584
0.533
0.557
0.584
0. 613
0.613
22.214
24.583
27.588
31.434
36.551
43.816
54.575
5.311
5.938
6.276
6.640
7.089
7.564
8.503
rov
0.8586
0.5587
0.5557
0.5528
0.5500
0.5945
0.4977
0.4678
0.4340
0.3949
0.3493
0.3179
rol
843.05
806.93
770.24
733.56
719.66
721.27
723.02
725.06
727.50
730.49
734.29
734.29
Output of the simulated 1-D loop analysis of 107R2 with 4-Eq model
time step no =
183
time = 183.000000 sec
number of newton Iterations number of Inner iterations =
1
0
0
total reactor power =
total heat transfer flow enthalpy rise =
time step size
inlet flow rate =
outlet flow rate o
0.210
-0.000
-0.000
=
0.100000+01 sec
0.000 kg/s
0.000 kg/s
maximum relative changes over the time step
0.000000
in pressure:
in mixture density: 0.000000
in mixture energy:
0.000000
ic
iz
P(bar)
void
% qual
hm
rom
T vap
T liq
T sat
1
1
1
1
1
1
1
1
1
1
1
2
3
4
s-5
6
7
8
9
10
1.00000
T7
0.97494
0.97460
.74217
1.02307
1.07187
1.12066
1.16946
1.21826
12
13
14
1.31586
1.31552
1.31519
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
1128251.
1128248.
1128248.
1128248.
1128248.
1128254.
1128260.
1128266.
1128272.
1128278.
1128284.
1128291.
1128291.
1128291.
1128292.
913454.
1002726.
1092001.
1181276.
1270551.
1359825.
1359823.
1359820.
1359817.
1359813.
1359810.
1128207.
1128251.
811. 34
811.33
811.33
811.33
811.33
811. 33
811.33
811. 33
811.33
811.33
811.33
811.33
811.33
811.33
811.33
850.63
834.44
818.04
801.50
784.85
768.16
768.16
768. 16
768.16
768. 16
768.16
811.34
811.34
863.15
863. 19
863.19
863.19
863. 19
863.19
863.19
863. 19
863.19
863.19
863. 19
863.19
863. 19
863.19
863.19
693.15
763.55
834.35
905.33
976.25
1046.91
1046.91
1046.91
1046.90
1046.90
1046.90
863.15
863.15
863.15
863.19
863.19
863.19
863.19
863.19
863. 19
863.19
863.19
863.19
863. 19
863.19
863.19
863.19
863. 19
693.15
763.55
834.35
905.33
976.25
1046.91
1046.91
1046.91
1046.90
1046.90
1046.90
863.15
863.15
1157.32
1154.55
1154.51
1154.47
1
1
1
1
1
1
1
1
1
1
1
1
1
1
.1
1
1
0.0000
0000
.o
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
6. 7
_15
14I486
16
17
18
19
20
21
22
23
24
25
26
27
28
1.26540
1.23131
1.21532
1.19965
1.18430
1.16926
1.15053
1.12778
1.10503
1.08227
1.05952
1.02A73
1.00000
-172-
1154.44
1159.85
1165.07
1170.10
1174.95
1179.65
1184.20
1188.61
1188.58
1188.55
1188.52
1184.05
1180.88
1179.37
1177.88
1176.40
1174.93
1173.09
1170.82
1168.51
1166.16
1163.77
1160.04
1157.32
vvz
vlz
0.070
0.070
0.070
0.070
0.070
0.070
0.070
0.070
0.070
0.070
0.070
0.070
0.070
0.070
0.070
0.067
0.068
0.069
0.071
0.072
0.074
0.074
0.074
0.074
0.074
0.074
0.070
0.070
0.070
0.070
0.070
0.070
0.070
0.070
0.070
0.070
0.070
0.070
0.070
0.070
0.070
0.070
0.067
0.068
0.069
0.071
0.072
0.074
0.074
0.074
0.074
0.074
0.074
0.070
rov
0.3396
0.2643
0.2643
0.2642
0.2641
0.2763
0.2885
0.3006
0.3127
0.3247
0.3367
0.3487
0.3486
0.3485
0.3484
0.3363
0.3279
0.3240
0.3201
0.3163
0.3126
0.3080
0.3024
0.2967
0.2911
0.2854
0.2767
0.3396
rol
811.34
811.33
811.33
811.33
811.33
811.33
811.33
811.33
811.33
811.33
81 1.33
811.33
811.33
811. 33
811.33
850.63
834.44
818.04
801.50
784.85
768.16
768. 16
768.16
768.16
768.16
768.16
811.34
811.34
Output of the actual 1-D loop analysis of 107R2 with 4-Eq model
time
step no
time = 143.199102 sec
= 2437
nunmber of newton iterations =
number of inner iterations =
total reactor power
total heat transfer =
flow enthalpy rise =
1
O
time step size = 0.585010-01 sec
0
0.210
-0.000
-0.000
inlet flow rate =
outlet flow rate =
0.000 kg/s
-0.000 kg/s
maximum relative changes over the time step
In pressure:
0.000000
in mixture density:
0.000000
In mixture energy:
0.000000
ic
iz
P(bar)
1 .34223
1 .34191
1 .32992
1. 9337
1.26128
1 .24993
1 .23059
1 .21524
1. 19990
1.18036
1. 15663
1 13291
1 10918
1.08545
1.04916
1.01961
1.00032
1.00000
void
% qual
hm
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
1128251.
1128339.
1128338.
1128333.
1128329.
1128328.
1128326.
1128324.
1128322.
1128320.
1128317.
1128314.
1128311.
1128308.
1128303.
1128299.
1128297.
1128298.
rom
811.34
811.33
811.33
811.33
811.33
811.33
811.33
811.33
811.33
811.33
811.33
811. 33
811.33
811.33
811.33
811.33
811.33
811.33
T vap
T liq
863.15
863.22
863. 22
863.22
863.22
863.22
863.22
863.22
863.22
863.22
863.22
863.22
863.22
863. 22
863. 22
863.22
863.22
863. 19
83. 15
863.22
863.22
863.22
863.22
863.22
863.22
863.22
863.22
863.22
863.22
863.22
863.22
863.22
863.22
863.22
863.22
863. 19
-173-
T sat
1190.94
1190.91
1189.86
1186.59
1183.67
1182.25
1180.81
1179.36
1177.90
1176.02
1173.69
1171.33
1168.93
1166.49
1162.67
t158.71
1157.36
1157.32
VVZ
0.000
-0.063
-0.063
-0.063
-0.063
-0.063
-0.063
-0.063
-0.063
-0.063
-0.063
-0.063
-0.063
-0.063
-0.063
-0.063
-0.063
vlz
0.000
-0.063
-0.063
-0.063
-0.063
-0.063
-0.063
-0.063
-0.063
-0.063
-0.063
-0.063
-0.063
-0.063
-0.063
-0.063
-0.063
rov
0.4558
0.3550
0.3521
0.3432
0.3353
0.3315
0.3277
0.3240
0.3202
0.3154
0.3095
0.3036
0.2977
0.2918
0.2828
0.2737
0.2706
0.3396
rol
8 11
.34
811. 33
811.33
811. 33
811.33
811.33
81 1.33
811.33
811 .33
811.33
811.33
811 .33
811.33
811.33
811.33
81 1.33
811.33
811.33
1.34085
1.34052
1.63682
1.63682
1.63682
1.63682
1.63682
1.63682
1.63682
1.91335
1.91335
1.91335
1.91335
1 .91335
1.91335
1.91335
1.00032
1.00000
1.33993
1.33961
1.63682
1.63682
1.63682
1.63682
1.63682
1.63682
1.91335
1 .91335
1.91335
S. 91335
11.91335
.91335
S.91335
1.00032
1.00000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000,
0.000
0.000
0.000
0.0000
0.0000
0.0000
0.0000
0.000()
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
o.oooo
0.000
0.000
0.000
0 000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
1128251.
1128338.
1128305.
1128305.
1128305.
1128305.
1128305.
1128305.
1128305.
1128364.
1128364.
1128364.
1128364.
1128364.
1128364.
1128364.
1128262.
1128298.
1128251.
1128335.
1128305.
1128305.
1128305.
1128305.
1128305.
1128305.
1128305.
1128364.
1128364.
1128364.
1128364.
1128364.
1128364.
1128364.
1128240.
1128287.
811. 34
811.33
81 1.34
811.34
811. 34
811. 34
8 11 .34
811. 34
811 .34
811. 33
811 .33
81.
33
811. 33
811.33
811 . 33
811. 33
811 .33
811.34
811.33
811.34
8li 34
811.34
811.34
Sii.34
811.34
81t.34
811. 33
8 11 .33
.33
811
811.33
811.33
811.33
811
.33
811.33
811 .34
8 1.34
863. 15
863.22
863. 17
863.17
863. 17
863. 17
863. 17
863. 17
863.17
863. 19
863. 19
863.19
863. 19
863. 19
863. 19
863.19
863.20
863. 19
863. 15 1190.82
863.22 1190.79
863. 17 1214.75
863. 17 1214.75
863. 17 1214.75
863. 17 1214.75
863. 17 1214.75
863. 17 1214.75
863. 17 1214.75
863. 19 1234.19
863. 19 1234.19
863. 19 1234. 19
863. 19 1234.19
863. 19 1234.19
863. 19 1234.19
863.19 1234.19
863.20 1157.36
863. 19 1157.32
863. 15
863.22
863.17
863. 17
863. 17
863. 17
863. 17
863. 17
863. 17
863. 19
863.19
863. 19
863. 19
863. 19
863. 19
863.19
863.18
863.18
863. 15
863.22
863. 17
863. 17
863. 17
863. 17
863. 17
863. 17
863. 17
863. 19
863. 19
863. 19
863. 19
863. 19
863. 19
863.19
863. 18
863.18
-174-
1190.74
1190.71
1214.75
1214.75
1214.75
1214.75
1214.75
1214.75
'1214. 75
1234. 19
1234. 19
1234. 19
1234. 19
1234. 19
1234. 19
1234.19
1157.36
1157.32
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
-0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
-0.000
0.000
0.000
0.000
0.000
0.000
0 000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.4554
0.3547
0.4264
0.4264
0.4264
0.4264
0.4264
0.4264
0.4264
0.4923
0.4923
0.4923
0.4923
0.4923
0.4923
0.4923
0.2706
0.3396
811 .34
811.33
811. 34
8 1.34
811.34
811 .34
811.34
811. 34
811 .34
811. 33
811 .33
811 .33
811.33
0.4551
0.3545
0.4264
0.4264
0.4264
0.4264
0. 4264
0.4264
0.4264
0.4923
0.4923
0.4923
0.4923
0.4923
0.4923
0.4923
0.2706
0.3396
81 1.34
811.33
811. 34
811 .33
811.33
811.33
811 .33
811.33
811.34
811.34
811. 34
8 11. 34
811.34
811.34
811.33
811.33
811.33
811.33
811.33
811.33
8 11. 33
811.34
811.34
.33901
33869
.37503
.2 8R05
25399
.23805
.22247
.20724
.19937
17388
15143
12898
10653
0P .108
S04975
.01276
00033
.00000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0 0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
1128251.
1128335.
1128332.
913457.
1012871.
1119288.
1211706.
1311125.
1410545.
1410552.
1410557.
1410558.
1410551.
1410538.
1128210.
1128206.
1128205.
1128252.
811.34
811.33
863. 15
811.33
850.63
863.22
693. 15
771 .58
850.47
929.52
1008.41
1086.84
1086.85
1086.85
1086.86
1086.85
1086.85
863. 15
863. 15
863.15
863. 15
832.58
814.29
795.83
777.27
758.68
758.68
758.67
758.67
758.67
758.68
811. 34
811. 34
811.34
811.34
863.22
-175-
863. 15
863. 22
863.22
1190.66
1190.63
693.15
1186.11
1182.99
1181.51
1180.05
1178.60
1177.18
1175.39
1173.18
1170.94
1168.66
1166.35
1162.73
1158.73
1157.36
1157.32
771.58
850.47
929.52
1008.41
1086.84
1086.85
1086.85
1086.86
1086.85
1086.85
863.15
863. 15
863.15
863. 15
1189.42
0.000
0.063
0.063
0.060
0.061
0.063
0.064
0.066
0.067
0.067
0.067
0.067
0.067
0.067
0.063
0.063
0.062
0.000
0.063
0.063
0.060
0.061
0.063
0.064
0.066
0.067
0.067
0.067
0.067
0.067
0.067
0.063
0.063
0.062
0.4547
0.3542
0.3509
0.3418
0.3335
0.3296
0.3257
0.3220
0.3183
0.3138
0.3082
0.3027
0.2971
0.2915
0.2830
0.2737
0.2706
0.3396
811.34
811 .33
811.33
850.63
832.58
814.29
795.83
777.27
758.68
758.68
758.67
758.67
758.67
758.68
811.34
811.34
811.34
811 .34
-9LT-
00+0000'0
00+0000"0
00+0000 0
00+0000 0
00+0000"0
00+0000"0
00+0000 0
00+0000 0
00+00000
00+0000"0
00+0000O
00+0000"0
00+0000 *0
00+0000"0
00+0000O0
00+0000"0
O0+000"O
00+0000"0
00+0000 0
00+0000"0
00+0000 0
00+0000 0
00+0000O0
00+0000"0
00+0000' 0
00+0000'0
00+(1000'0
00+0000" 0
00+0000"0
00+0000 0
00+0000 0
00+0000"0
00+0000"0
00+0000*0
00+0000"0
00+0000
00+0000 0
00+0000"0
00+0000"0
00+0000"0
00+0000"0
00+0000'0
O0O000"O
AtA
00+0000*0
00.0000"0
00+0000"0
00+0000 0
PO-OOLB't00+0000"0
00+0000" 0
00+0000*0
00+0000"0
00+00000
00+0000"0
00+0000 0
00+0000"0
00+0000'0
00+0000 0
00+0000"0
00+0000O
00+0000 0
90-0080"s00+0000*0
00+0000*0 00+0000"0
00+0000"0
00+0000 0
00+0000,0
00+0000 0
00+0000 0 00+0000*0
00+0000"0
00+0000 0
00+0000 ,0 00+0000 0
00+0000 0 00+0000 *0
00+0000 O 00+0000 "0
00+0000 0 00+0000"0
00+0000"0
00+00000
00+0000"0
00+0000O0 00+0000"0
00+0000O0 00+0000'0
00+0000
cOOsOO'O
O-OOPL'
00+0000:0
00+0000 0
00+0000,0
00+0000,0
00+0000 0
00+00000
00+00000
00+ 0000 0
00+0000'0
00+UOOO *0
00+0000"0
00+0000 0
00+0000'0
00+0000'0
00+0000' 0
AAA
XtA
00+GOOO*0'
00+0000-0
00+0000*00+0000"0
00+00000'0
00+0000"O
00+0000"0
00+0000'0
00+00000
00+ 0000 *0
00+00000
00+ 0000 0
00+0000,0
00+0000,0
00+0000'0
c0-out~ *c
XAA
)I
ZI
00+0000,0
00+0000"0
00+0000 0
00+0000"0
00+0000"0
00+0000"0
00+0000 0
00+0000,0
00+0000"0
001-0000 O0
00+0000,0
00+0000-0
00+0000-0
00+0000"0
00+0000"0
00. (10000*0
00+00000
00+00000"
00+0000 0
00+0000"0
00+0000*0
00+0000"0
00+0000"0
00+0000,0
00+(000 0
00+0000 0
00+0000 0
00+0000O0
00+0000"0
00+0000 0
00100000
00t 0000 0
00+0000,0
00+0000"0
00+0000,0
00+0000"0
00+0000*0
00+0000"0
00+0000"0
00+0000"0
00+0000"0
00+0000"0
000000"0
00+0000,0
00+0000,0
00i0000"0
00+0000O0
00+0000'0
00+0000*0 00+0000-0
00+0000 0
00+0000*0
00+0000"0 00+0000*0
00+00000 00+0000
00+0000"0 00+0000"0
00+0000"0 00+0000'0
00+0000'0 100+0000"0
00+0000"0 S00+0000*0
OO+000OOO
00+0000"0
00+0000"0 00+0000"0
00+0000'0 00+0000"0
00+0000"0
00+0000 0
00+0000*0
00+0000 0
00+0000"0 00+0000*0
00+00000" 00+0000 0
00+0000'0
00+0000*0 00+0000-0
AlA
AAA
00+0000 0
00+0000 O
00+0000'0
00+0000 0
00+0000,0
00+0000 O
0040000"0
00+0000*0
00+0000O0
00+ 0000 0
00+0000*0
00+0000 0)
00.00000
00*00000
00+0000 0
CO-OUCI%A
XIA
00+0000'0
00+0000-0
00+0000,0
00+0000 0
00+00000
00+0000 0
00+0000"0
00+0000*0
00+0000*0
00+0000
00+0000*0
00+ 0000 0
0040000 O
00+000O
00 0000,0
00 0000 0
00+0000*0
00+O0000O
00+0000*0
00+0000O
00+0000 0
000000"0
00+0000"0
00+0000*0
00+0000'0
00+0000"0
00+0000 0
Output of the simulated 1-D loop analysis with vapor density x200 at boiling inception
time step size = 0.100000+00 sec
time =
3.000000 sec
29
time step no =
number of newton Iterations =
2
number of inner iterations *
1
1
O
total reactor power total heat transfer =
flow enthalpy rise =
inlet flow rate =
outlet flow rate =
0.9 10
0.903
1.823
0.000 kg/s
0.002 kg/s
maximum relative changes over
in pressure:
in mixture density:
in mixture energy:
ic
iz
P(bar)
void
% qual
hm
1.00000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0492
0.0848
0.1141
0. 1082
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.000
0. 000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
1128298.
1128252.
1128248.
1128248.
1128248.
1128254.
1128260.
1128266.
1128273.
1128279.
1128285.
1128291.
1128291.
1128248.
1123744.
913455.
1432427.
1551804.
1567326.
1580402.
1574938.
1280190.
1160858.
1133197.
1128820.
1128308.
1128209.
1128253.
0.97545
0.97547
0.97550
0.97552
1.02468
1.07384
1 .12299
1.17215
1.22131
1.27046
1.31962
1.31964
1.31967
1.31969
1.27054
1.23671
1 .22230
1.20852
1. 19516
1. 18134
1. 16354
1. 13924
1. 11431
1 .08924
1.06416
1.02581
1.00000
0.000
0.000
2.925
2.681
1. 106
1.030
0.000
0.000
0.000
0.000
0.000
0.000
rom
T vap
811 .33
811.33
811 . 33
811. 33
811.33
811. 33
863.19
863. 19
863.19
863.19
863. 19
863. 19
811.33
863.19
811. 33
863.19
811.33
863. 19
811.33
863.19
811.33
863. 19
811. 33
863.19
863.19
81 1.33
863.15
811. 34
812.18
859.57
850.63
693.15
754.59 1104.01
703.39 1180.03
679.72 1178.73
660.21 1177.45
664.43 1176.11
783.05 983.90
889.10
805.29
867.11
810.42
863.63
811.23
863.23
811.32
863. 15
811.34
863.15
811.34
-177-
T liq
the time step
0.001371
0.039481
0.018382
T sat
863. 19
863. 19
863. 19
863. 19
863.19
'863. 19
863. 19
1157.32
1154.57
1154. 57
1154.57
1154.58
1160.03
1165.28
863.19 1170.33
863.19 1175.22
863. 19 1179.94
863.19 1184.51
863. 19 1188.94
863. 19 1188.95
863. 15 1188.95
859.57 1188.95
693.15 1184.52
1104.01 1181.39
1180.03 1180.03
1178.73 1178,73
1177.45 1177.45
1176.11 1176 11
983.90 1174.37
889.10 1171 .97
867. 11 1169.45
863.63 1166.88
863.23 1164.26
863. 15 1160. 15
863. 15 1157.32
vvz
vlz
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.005
0.012
0.425
0.403
0. 199
0.263
0.246
0.246
0.246
0.246
0.246
0.246
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.005
0.012
0.065
0. 119
0.198
0.263
0.246
0.246
0.246
0.246
0.246
0.246
rov
67.9204
52.8764
52.8777
52.8790
52. 8802
55. 3411
57.7922
60.2340
62.6668
65.0910
67.5070
69.9150
69.9163
69.9175
69.9188
67.5107
65.8488
65.1401
64.4613
63.8028
63.1207
62.2413
61.0391
59.8031
58.5585
57.3107
55.3977
67.9226
rol
811.33
811.33
811.33
811.33
811.33
811.33
811.33
811.33
811.33
811.33
811.33
811.33
811.33
811.34
812. 18
850.63
754.59
736.41
736.72
737.03
737.35
783.05
805.29
810.42
811.23
811.32
811.34
811.34
r
c
Sr
with vapor density xl00 at boiling inception
loop analysis
___
Output of the simulated 1-D
numbmbr of
newton
number of
inner
time
27
step no =
iterations
Iterations
total reactor power =
total heat transfer =
flow enthalpy rise =
=
u
=
3.336324 sec
time step size = 0.838680-01 sec
2
2
0
0
inlet flow rate =
outlet flow rate =
0.910
0.674
5.728
-0.001 kg/s
0.005 kg/s
maximum relative changes over the time step
0.052486
in pressure:
in mixture density: 0.263107
0.115685
in mixture energy:
Ic
iz
P(bar)
1.00000
0.97620
0.97774
0.97927
0.98080
1.03147
1.08213
1. 13280
1.18346
1.23413
1.28479
1. 33546
1.33699
1.33853
1.34007
1.29023
. 26036
1.25770
1.25186
1.23878
1 .22303
1.20083
1.17281
1. 14247
1. 11143
1.08016
1.03225
1.00000
void
% qua)
0.0000 0.000
0.0000 0.000
0.0000 0.000
0.0000 0.000
0.0000 0.000
0.0000 0.000
0.0000 0.000
0.0000
0.000
0.000
0.0000
0.0000 0.000
0.0000 0.000
0.0000 0.000
0.0000 0.000
0.0000 0.000
0.0000 0.000
0.0000 0.000
0.0000 0.000
0.8209 23.083
5.120
0.4802
0.3837
3. 2 10
2.802
0. 3626
0.0000 0.000
0.0000 0.000
0.0000 0.000
0.0000 0.000
0.0000 0.000
0.0000 0.000
0.0000
hm
1128302.
1128252.
1128249.
1128249.
1128249.
1128255.
1128261.
1128268.
1128274.
1128280.
1128286.
1128292.
1128254.
1127040.
1103926.
913845.
1336537.
2387063.
1731457.
1665481.
1650778.
1469082.
1284700.
1180667.
1141395.
1130786.
1128223.
1128266.
rom
T vap
863.19
863. 19
863. 19
811.33
863.19
811.33
863.19
811.33
863.19
811.33
811.33
863.19
863.19
811.33
863.19
811.33
863.19
811.33
863.19
811.33
863. 19
811.33
863.16
811.34
862. 19
8 11. 57
843.82
815.84
693.45
850.56
772.52 1028.50
159.19 1183.34
398.41 1182.80
466.30 1181.58
481. 17 1180. 10
747.75 1132.69
987.48
782.21
904.85
801.61
873.63
808.90
865.20
810.87
863.16
811.34
863.16
811.34
811.33
811.33
-178-
T liq
T sat
863.19
863.19
863.19
863. 19
863.19
863.19
863. 19
863.19
863.19
863.19
863. 19
863.19
863. 16
862.19
843.82
693.45
1028.50
1183.34
1182.80
1181.58
1180.10
1132.69
987.48
1157.32
1154.65
1154.83
1155.00
1155. 17
1160.77
1166.14
1171.32
1176.32
1181.15
1185.82
1190.34
1190.48
1190.61
1190.75
1186.31
1183.58
1183.34
1182.80
1181.58
1180.10
1177.99
1175.28
1172.29
1169.16
1165.94
1160.85
1157.32
904.85
873.63
865.20
863.16
863.16
vvz
-0.081
-0.081
-0.081
-0.081
-0.081
-0.081
-0.081
-0.081
-0.081
-0.081
-0.081
-0.081
-0.081
-0.081
-0.081
-0.089
0.318
0.390
0.659
0.857
1.049
0.675
0.674
0.674
0.674
0.674
0.674
vlz
-0.081
-0.081
-0.081
-0.081
-0.081
-0.081
-0.081
-0.081
-0.081
-0.081
-0.081
-0.081
-0.081
-0.081
-0.081
-0.089
-0.410
0.271
0.510
0.720
0.916
0.675
0.674
0.674
0.674
0.674
0.674
rov
33.9601
26.4571
26.4956
26.5341
26.5725
27.8401
29.1025
30.3600
31.6127
32.8610
34.1049
35.3447
35.382 1
35.4196
35.4574
34.2381
33.5056
33.4403
33.2967
32.9755
32.5879
32.0410
31.3497
30.5995
29.8303
29.0534
27.8597
33.9610
rol
8 11. 33
811.33
811.33
811.33
811.33
811.33
811.33
8 11. 33
811.33
811.33
811. 33
811.33
811.34
811.57
815.84
850.56
772.52
735.62
735.75
736.04
736.39
747.75
782.21
801.61
808.90
810.87
811.34
811.34